Research ArticleVague Filters of Residuated Lattices

Shokoofeh Ghorbani

Department of Mathematics 22 Bahman Boulevard Kerman 76169-133 Iran

Correspondence should be addressed to Shokoofeh Ghorbani shghorbaniukacir

Received 2 May 2014 Revised 7 August 2014 Accepted 13 August 2014 Published 10 September 2014

Academic Editor Hong J Lai

Copyright copy 2014 Shokoofeh GhorbaniThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Notions of vague filters subpositive implicative vague filters and Boolean vague filters of a residuated lattice are introduced andsome related properties are investigated The characterizations of (subpositive implicative Boolean) vague filters is obtained Weprove that the set of all vague filters of a residuated lattice forms a complete lattice andwe find its distributive sublatticesThe relationamong subpositive implicative vague filters and Boolean vague filters are obtained and it is proved that subpositive implicative vaguefilters are equivalent to Boolean vague filters

1 Introduction

In the classical set there are only two possibilities for anyelements in or not in the set Hence the values of elementsin a set are only one of 0 and 1 Therefore this theory cannothandle the data with ambiguity and uncertainty

Zadeh introduced fuzzy set theory in 1965 [1] to handlesuch ambiguity and uncertainty by generalizing the notionof membership in a set In a fuzzy set 119860 each element 119909 isassociated with a point-value 120583

119860(119909) selected from the unit

interval [0 1] which is termed the grade of membership inthe set This membership degree contains the evidences forboth supporting and opposing 119909

A number of generalizations of Zadehrsquos fuzzy set theoryare intuitionistic fuzzy theory L-fuzzy theory and vaguetheory Gau and Buehrer proposed the concept of vagueset in 1993 [2] by replacing the value of an element in aset with a subinterval of [0 1] Namely a true membershipfunction 119905V(119909) and a false-membership function 119891V(119909) areused to describe the boundaries of membership degreeThese two boundaries form a subinterval [119905V(119909) 1 minus 119891V(119909)]

of [0 1] The vague set theory improves description of theobjective real world becoming a promising tool to deal withinexact uncertain or vague knowledge Many researchershave applied this theory to many situations such as fuzzycontrol decision-making knowledge discovery and faultdiagnosis Recently in [3] Jun andPark introduced the notion

of vague ideal in pseudo MV-algebras and Broumand Saeid[4] introduced the notion of vague BCKBCI-algebras

The concept of residuated latticeswas introduced byWardand Dilworth [5] as a generalization of the structure of theset of ideals of a ring These algebras are a common structureamong algebras associated with logical systems (see [6ndash9])The residuated lattices have interesting algebraic and logicalproperties The main example of residuated lattices relatedto logic is and BL-algebras A basic logic algebra (BL-algebrafor short) is an important class of logical algebras introducedby Hajek [10] in order to provide an algebraic proof of thecompleteness of ldquoBasic Logicrdquo (BL for short) Continuous t-norm based fuzzy logics have been widely applied to fuzzymathematics artificial intelligence and other areasThe filtertheory plays an important role in studying these logicalalgebras and many authors discussed the notion of filters ofthese logical algebras (see [11 12]) From a logical point ofview a filter corresponds to a set of provable formulas

In this paper the concept of vague sets is applied to resid-uated lattices In Section 2 some basic definitions and resultsare explained In Section 3 we introduce the notion of vaguefilters of a residuated lattice and investigate some relatedproperties Also we obtain the characterizations of vaguefilters In Section 4 the smallest vague filters containing agiven vague set are established and we obtain the distributivesublattice of the set of all vague filters of a residuated latticeIn Section 5 notions of vague filters subpositive implicative

Hindawi Publishing CorporationJournal of Discrete MathematicsVolume 2014 Article ID 120342 9 pageshttpdxdoiorg1011552014120342

2 Journal of Discrete Mathematics

vague filters and Boolean vague filters of a residuated latticeare introduced and some related properties are obtained

2 Preliminaries

We recall some definitions and theorems which will beneeded in this paper

Definition 1 (see [6]) A residuated lattice is an algebraicstructure (119871 and or lowast rarr 999492999484 0 1) such that

(1) (119871 and or 0 1) is a bounded lattice with the least ele-ment 0 and the greatest element 1

(2) (119871 lowast 1) is amonoid that islowast is associative and119909lowast1 =

1 lowast 119909 = 119909

(3) 119909 lowast 119910 le 119911 if and only if 119909 le 119910 rarr 119911 if and only if119910 le 119909 999492999484 119911 for all 119909 119910 119911 isin 119871

In the rest of this paper we denote the residuated lattice(119871 and or lowast rarr 999492999484 0 1) by 119871 not119909 = 119909 rarr 0 and sim 119909 = 119909 999492999484 0

Proposition 2 (see [6 9]) Let 119871 be a residuated lattice Thenwe have the following properties for all 119909 119910 119911 isin 119871

(1) 119909 rarr 119909 = 119909 999492999484 119909 = 1

(2) 1 rarr 119909 = 1 999492999484 119909 = 119909

(3) 119909 rarr (119910 999492999484 119911) = 119910 999492999484 (119909 rarr 119911)

(4) 119909 le 119910 if and only if 119909 rarr 119910 = 1 if and only if 119909 999492999484 119910 =

1

(5) if 119909 le 119910 then 119909 lowast 119911 le 119910 lowast 119911 119911 lowast 119909 le 119911 lowast 119910

(6) if 119909 le 119910 then 119911 rarr 119909 le 119911 rarr 119910 119911 999492999484 119909 le 119911 999492999484 119910

(7) if 119909 le 119910 then 119910 rarr 119911 le 119909 rarr 119911 119910 999492999484 119911 le 119909 999492999484 119911

(8) 119909 le (119909 999492999484 119910) rarr 119910 119909 le (119909 rarr 119910) 999492999484 119910

(9) 119910 rarr 119911 le (119909 rarr 119910) 999492999484 (119909 rarr 119911) 119910 999492999484 119911 le (119909 999492999484

119910) 999492999484 (119909 999492999484 119911)

(10) 119909 rarr (119910 999492999484 (119909 lowast 119910)) = 1 119910 999492999484 (119909 rarr (119909 lowast 119910)) = 1

Definition 3 (see [6 9]) Let 119865 be a nonempty subset of aresiduated lattice 119871 119865 is called a filter if

(F1) 119909 119910 isin 119865 imply 119909 lowast 119910 isin 119865

(F2) 119909 le 119910 and 119909 isin 119865 imply 119910 isin 119865

Theorem 4 (see [6 9]) A nonempty subset 119865 of a residuatedlattice 119871 is a filter if and only if

(F3) 1 isin 119865 and 119909 119909 rarr 119910 isin 119865 imply 119910 isin 119865 or

(F31015840) 1 isin 119865 and 119909 119909 999492999484 119910 isin 119865 imply 119910 isin 119865

Definition 5 (see [11]) Let 119865 be a nonempty subset of aresiduated lattice 119871 119865 is called a subpositive implicative filterof 119871 if

(SPIF1) 1 isin 119865(SPIF2) ((119909 rarr 119910) lowast 119911) 999492999484 ((119910 999492999484 119909) rarr 119909) isin 119865

and 119911 isin 119865 imply (119909 rarr 119910) 999492999484 119910 isin 119865 for any119909 119910 119911 isin 119871

(SPIF3) (119911 lowast (119909 999492999484 119910)) rarr ((119910 rarr 119909) 999492999484 119909) isin 119865

and 119911 isin 119865 imply (119909 999492999484 119910) rarr 119910 isin 119865 for any119909 119910 119911 isin 119871

Proposition 6 (see [11]) Let119865 be a filter of119871119865 is a subpositiveimplicative filter if and only if

(SPIF4) (119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909) isin 119865 imply(119909 rarr 119910) 999492999484 119910 isin 119865 for any 119909 119910 isin 119871

(SPIF5) (119909 999492999484 119910) rarr ((119910 rarr 119909) 999492999484 119909) isin 119865 imply(119909 999492999484 119910) rarr 119910 isin 119865 for any 119909 119910 isin 119871

Definition 7 (see [1]) Let119883 be a nonempty set A fuzzy set in119883 is a mapping 120583 119883 rarr [0 1]

Proposition 8 (see [12]) Let 119871 be a residuated lattice A fuzzyset 120583 is called a fuzzy filter in 119871 if it satisfies

(fF1) 119909 le 119910 imply 120583(119909) le 120583(119910)

( fF2) min120583(119909) 120583(119910) le 120583(119909 lowast 119910)

for all 119909 119910 isin 119871

Definition 9 (see [13]) A vague set 119860 in the universe ofdiscourse 119883 is characterized by two membership functionsgiven by

(i) a true membership function 119905119860

119883 rarr [0 1]

(ii) a false membership function 119891119860

119883 rarr [0 1]

where 119905119860(119909) is a lower bound on the grade of membership of

119883 derived from the ldquoevidence for 119909rdquo 119891119860(119909) is a lower bound

on the negation of 119909 derived from the ldquoevidence against 119909rdquoand 119905119860(119909) + 119891

119860(119909) le 1

Thus the grade of membership of 119909 in the vague set 119860

is bounded by a subinterval [119905119860(119909) 1 minus 119891

119860(119909)] of [0 1] This

indicates that if the actual grade of membership of 119909 is 120583(119909)then 119905

119860(119909) le 120583(119909) le 1 minus 119891

119860(119909) The vague set 119860 is written

as 119860 = ⟨119909 [119905119860(119909) 1 minus 119891

119860(119909)]⟩ 119909 isin 119883 where the interval

[119905119860(119909) 1 minus 119891

119860(119909)] is called the vague value of 119909 in 119860 denoted

by 119881119860(119909)

Definition 10 (see [13]) A vague set 119860 of a set 119883 is called

(1) the zero vague set of 119883 if 119905119860(119909) = 0 and 119891

119860(119909) = 1 for

all 119909 isin 119883

(2) the unit vague set of 119883 if 119905119860(119909) = 1 and 119891

119860(119909) = 0 for

all 119909 isin 119883

(3) the 120572-vague set of 119883 if 119905119860(119909) = 120572 and 119891

119860(119909) = 1 minus 120572

for all 119909 isin 119883 where 120572 isin (0 1)

Let 119868[0 1] denote the family of all closed subintervals of[0 1] Nowwedefine the refinedminimum (briefly imin) and

Journal of Discrete Mathematics 3

an order le on elements 1198681= [1198861 1198871] and 119868

2= [1198862 1198872] of 119868[0 1]

asimin (119868

1 1198682) = [min 119886

1 1198862 min 119887

1 1198872]

1198681le 1198682

iff 1198861le 1198862 1198871le 1198872

(1)

Similarly we can define ge = and imax Then the concept ofimin and imax could be extended to define iinf and isup ofinfinite number of elements of 119868[0 1] It is a known fact that119871 = 119868[0 1] iinf isup le is a lattice with universal boundszero vague set and unit vague set For 120572 120573 isin [0 1] we nowdefine (120572 120573)-cut and 120572-cut of a vague set

Definition 11 (see [13]) Let 119860 be a vague set of a universe 119883

with the true-membership function 119905119860and false-membership

function119891119860The (120572 120573)-cut of the vague set119860 is a crisp subset

119860(120572120573)

of the set 119883 given by

119860(120572120573)

= 119909 isin 119883 119881119860 (119909) ge [120572 120573]

where 120572 le 120573

(2)

Clearly 119860(00)

= 119883 The (120572 120573)-cuts are also called vague-cutsof the vague set 119860

Definition 12 (see [13]) The 120572-cut of the vague set119860 is a crispsubset 119860

120572of the set119883 given by 119860

120572= 119860(120572120572)

Equivalently wecan define the 120572-cut as 119860

120572= 119909 isin 119883 119905

119860(119909) ge 120572

3 Vague Filters of Residuated Lattices

In this section we define the notion of vague filters ofresiduated lattices and obtain some related results

Definition 13 A vague set 119860 of a residuated lattice 119871 is calleda vague filter of 119871 if the following conditions hold

(VF1) if 119909 le 119910 then 119881119860(119909) le 119881

119860(119910) for all 119909 119910 isin 119871

(VF2) imin119881119860(119909) 119881119860(119910) le 119881

119860(119909lowast119910) for all 119909 119910 isin

119871

That is(1) 119909 le 119910 implies 119905

119860(119909) le 119905

119860(119910) and 1minus119891

119860(119909) le 1minus119891

119860(119910)

for all 119909 119910 isin 119871(2) min119905

119860(119909) 119905119860(119910) le 119905

119860(119909 lowast 119910) and min1 minus 119891

119860(119909) 1 minus

119891119860(119910) le 1 minus 119891

119860(119909 lowast 119910) for all 119909 119910 isin 119871

In the following proposition we will show that the vaguefilters of a residuated lattice exist

Proposition 14 Unit vague set and120572-vague set of a residuatedlattice 119871 are vague filters of 119871

Proof Let 119860 be a 120572-vague set of 119871 It is clear that 119860 satisfies(VF1) For 119909 119910 isin 119871 we have

119905119860(119909 lowast 119910) = 120572 = min 120572 120572 = min 119905

119860 (119909) 119905119860 (119910)

1 minus 119891119860(119909 lowast 119910)

= 120572 = min 120572 120572 = min 1 minus 119891119860 (119909) 1 minus 119891

119860(119910)

(3)

Thus imin119881119860(119909) 119881119860(119910) le 119881

119860(119909 lowast 119910) for all 119909 119910 isin 119871 Hence

it is a vague filter of 119871 The proof of the other cases is similar

Lemma 15 Let 119860 be a vague set of a residuated lattice 119871 suchthat it satisfies (VF1) Then the following are equivalent

(VF4) iminVA(x)VA(x rarr y) le VA(y)(VF5) iminVA(x)VA(x 999492999484 y) le VA(y)

for all 119909 119910 isin 119871

Proof Suppose that 119860 satisfies (VF4) We have 119909 le (119909 999492999484

119910) rarr 119910 By (VF1) 119881119860(119909) le 119881

119860((119909 999492999484 119910) rarr 119910) We get that

imin 119881119860 (119909) 119881119860 (119909 999492999484 119910)

le imin 119881119860((119909 999492999484 119910) 997888rarr 119910) 119881

119860(119909 999492999484 119910)

le 119881119860(119910)

(4)

Conversely if119860 satisfies (VF5) then the inequality 119909 le (119909 rarr

119910) 999492999484 119910 analogously entails (VF4)

Theorem 16 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter if and only if it satisfies (VF4) and

(VF6) 119881119860(119909) le 119881

119860(1) for all 119909 isin 119871

Proof Let 119860 be a vague filter of 119871 Since 119909 le 1 then 119881119860(119909) le

119881119860(1) for all 119909 isin 119871 by (VF1) We have 119909 lowast (119909 rarr 119910) le 119910 By

(VF1) and (VF2) we get

imin 119881119860 (119909) 119881119860 (119909 997888rarr 119910)

le 119881119860(119909 lowast (119909 997888rarr 119910)) le 119881

119860(119910)

(5)

Hence (VF4) holdConversely let 119860 satisfy (VF4) and (VF6) Since 119909 le 119910

then 119909 rarr 119910 = 1

119881119860(119910) ge imin 119881

119860(119909 997888rarr 119910) 119881

119860 (119909)

= imin 119881119860 (1) 119881119860 (119909) = 119881

119860 (119909)

(6)

Hence 119881119860

is order-preserving and (VF1) holds ByProposition 2 part (10) 119909 rarr (119910 999492999484 (119909 lowast 119910)) = 1 By(VF4) we have

119881119860(119910 999492999484 (119909 lowast 119910))

ge imin 119881119860 (119909) 119881119860 (119909 997888rarr (119910 999492999484 (119909 lowast 119910))) = 119881

119860 (119909)

(7)

By Lemma 15

119881119860(119909 lowast 119910) ge imin 119881

119860(119910 999492999484 (119909 lowast 119910)) 119881

119860(119910)

ge imin 119881119860 (119909) 119881119860 (119910)

(8)

Hence (VF2) holds

Theorem 17 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter if and only if it satisfies (VF5) and(VF6)

4 Journal of Discrete Mathematics

Definition 18 A vague set 119860 of a residuated lattice 119871 is calleda vague lattice filter of 119871 if it satisfies (VF1) and

(VLF2) imin119881119860(119909) 119881119860(119910) le 119881

119860(119909 and 119910) for all

119909 119910 isin 119871

Proposition 19 Let 119860 be a vague filter of a residuated lattice119871 then 119860 is a vague lattice filter

In the following example we will show that the converseof Proposition 19 may not be true in general

Example 20 Let 0 119886 119887 119888 1 such that 0 lt 119886 lt 119887 119888 lt 1where 119887 and 119888 are incomparable Define the operations lowast rarr and 999492999484 as follows

lowast 0 119886 119887 119888 1

0 0 0 0 0 0

119886 0 0 0 119886 119886

119887 0 119886 119887 119886 119887

119888 0 0 0 119888 119888

1 0 119886 119887 119888 1

997888rarr 0 119886 119887 119888 1

0 1 1 1 1 1

119886 119888 1 1 1 1

119887 119888 119888 1 119888 1

119888 0 119887 119887 1 1

1 0 119886 119887 119888 1

999492999484 0 119886 119887 119888 1

0 1 1 1 1 1

119886 119887 1 1 1 1

119887 0 119888 1 119888 1

119888 119887 119887 119887 1 1

1 0 119886 119887 119888 1

(9)

Then 119871 is a residuated lattice Let 119860 be the vague set definedas follows

119860 = ⟨0 [01 02]⟩ ⟨119886 [03 04]⟩ ⟨119887 [03 04]⟩

⟨119888 [03 08]⟩ ⟨0 [05 09]⟩

(10)

It is routine to verify that119860 is a vague lattice filter but it is nota vague filter of 119871

Let 119861 be the vague set defined as follows

119861 = ⟨0 [02 05]⟩ ⟨119886 [02 05]⟩ ⟨119887 [02 05]⟩

⟨119888 [04 07]⟩ ⟨0 [07 07]⟩

(11)

Then 119861 is a vague filter of 119871

In the following theorems we will study the relationshipbetween filters and vague filters of a residuated lattice

Theorem 21 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter of 119871 if and only if for all 120572 120573 isin [0 1]the set 119860

(120572120573)is either empty or a filter of 119871 where 120572 le 120573

Proof Suppose that 119860 is a vague filter of 119871 and 120572 120573 isin [0 1]Let 119860

(120572120573)= 0 We will show that 119860

(120572120573)is a filter of 119871

(1) Suppose that 119909 le 119910 and 119909 isin 119860(120572120573)

By (VF1)119881119860(119910) ge

119881119860(119909) ge [120572 120573] Therefor 119910 isin 119860

(120572120573)

(2) Suppose that 119909 119910 isin 119860(120572120573)

Then 119881119860(119909) 119881119860(119910) ge

[120572 120573] By (VF2) we have 119881119860(119909 lowast 119910) ge

imin119881119860(119909) 119881119860(119910) ge [120572 120573] Hence 119909 lowast 119910 isin 119860

(120572120573)

and then 119860(120572120573)

is a filter of 119871

Conversely suppose that for all 120572 120573 isin [0 1] the sets 119860(120572120573)

is either empty or a filter of 119871 Let 119905119860(119909) = 120572

1 119905119860(119910) = 120572

2

1 minus 119891119860(119909) = 120573

1 and 1 minus 119891

119860(119910) = 120573

2 Put 120572 = min120572

1 1205722

and 120573 = min1 minus 1205731 1 minus 120573

2 Then 119905

119860(119909) 119905119860(119910) ge 120572 and 1 minus

119891119860(119909) 1 minus 119891

119860(119910) ge 120573 Hence 119881

119860(119910) 119881119860(119909) ge [120572 120573] that is

119909 119910 isin 119860(120572120573)

So 119860(120572120573)

= 0 and by assumption 119860(120572120573)

is afilter of 119871 We obtain that 119909 lowast 119910 isin 119860

(120572120573) that is

119881119860(119909 lowast 119910) ge [120572 120573]

= [min 1205721 1205722 min 1 minus 120573

1 1 minus 120573

2]

= imin [119905119860 (119909) 1 minus 119891

119860 (119909)] [119905119860 (119910) 1 minus 119891119860(119910)]

= imin 119881119860 (119909) 119881119860 (119910)

(12)

Hence (VF2) holdsSuppose that 119909 le 119910 119905

119860(119909) = 120572 and 1 minus 119891

119860(119909) = 120573 Hence

119909 isin 119860(120572120573)

By assumption 119860(120572120573)

is a filter We get that 119910 isin

119860(120572120573)

that is 119881119860(119910) ge [120572 120573] = [119905

119860(119909) 1 minus 119891

119860(119909)] = 119881

119860(119909)

Hence 119860 is a vague filter of 119871

The filters like119860(120572120573)

are also called vague-cuts filters of 119871

Corollary 22 Let 119860 be a vague filter of a residuated lattice 119871Then for all 120572 isin [0 1] the set 119860

120572is either empty or a filter of

119871

Corollary 23 Any filter 119865 of a residuated lattice 119871 is a vague-cut filter of some vague filter of 119871

Proof For all 119909 isin 119871 define

1 minus 119891119860 (119909) = 119905

119860 (119909) = 1 if 119909 isin 119865

120572 otherwise(13)

Hence we have 119881119860(119909) = [1 1] if 119909 isin 119865 and 119881

119860(119909) = [120572 120572]

otherwise where 120572 isin (0 1) It is clear that 119865 = 119860(11)

Let119909 119910 isin 119871 If 119909 119910 isin 119865 then 119909 lowast 119910 isin 119865 We have 119881

119860(119909 lowast 119910) =

[1 1] = imin119881119860(119909) 119881119860(119910) If one of 119909 or 119910 does not belong

to 119865 then one of119881119860(119909) and119881

119860(119910) is equal to [120572 120572]Therefore

119881119860(119909lowast119910) ge [120572 120572] = imin119881

119860(119909) 119881119860(119910) Suppose that 119909 le 119910

If 119909 isin 119865 then 119910 isin 119865 Hence 119881119860(119909) = 119881

119860(119910) If 119909 notin 119865 then

119881119860(119909) = [120572 120572] We have 119881

119860(119909) le 119881

119860(119910) Hence 119860 is a vague

filter of 119871

In the following proposition we will show that vaguefilters of a residuated lattice are a generalization of fuzzyfilters

Proposition 24 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter of 119871 if and only if the fuzzy sets 119905

119860and

1 minus 119891119860are fuzzy filters in 119871

Journal of Discrete Mathematics 5

4 Lattice of Vague Filters

Let 119860 and 119861 be vague sets of a residuated lattice 119871 If 119881119860(119909) le

119881119861(119909) for all 119909 isin 119871 then we say 119861 containing119860 and denote it

by 119860 ⪯ 119861The unite vague set of a residuated lattice 119871 is a vague

filter of 119871 containing any vague filter of 119871 For a vague setwe can define the intersection of two vague sets 119860 and 119861 of aresiduated lattice 119871 by 119881

119860cap119861(119909) = imin119881

119860(119909) 119881119861(119909) for all

119909 isin 119871 It is easy to prove that the intersection of any family ofvague filters of 119871 is a vague filter of 119871 Hence we can define avague filter generated by a vague set as follows

Definition 25 Let 119860 be a vague set of a residuated lattice 119871A vague filter 119861 is called a vague filter generated by 119860 if itsatisfies

(i) 119860 ⪯ 119861(ii) 119860 ⪯ 119862 that implies 119861 ⪯ 119862 for all vague filters 119862 of 119871

The vague filter generated by 119860 is denoted by ⟨119860⟩

Example 26 Consider the vague lattice filter119860 in Example 20which is not a vague filter of 119871Then the vague filter generatedby 119860 is

⟨119860⟩ = ⟨0 [03 04]⟩ ⟨119886 [03 04]⟩

⟨119887 [03 04]⟩ ⟨119888 [03 08]⟩ ⟨0 [05 09]⟩

(14)

Theorem 27 Let 119860 be a vague set of a residuated lattice 119871Then the vague value of 119909 in ⟨119860⟩ is

119881⟨119860⟩ (119909) = isup imin VA (a1) VA (a2) VA (an)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

(15)

for all 119909 isin 119871

Proof First we will prove that ⟨119860⟩ is a vague set of 119871

119905⟨119860⟩ (119909)

= sup min 119905119860(1198861) 119905119860(1198862) 119905

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

le sup min 1 minus 119891119860(1198861) 1 minus 119891

119860(1198862) 1 minus 119891

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

= sup 1 minus max 119891119860(1198861) 119891119860(1198862) 119891

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

= 1 minus inf max 119891119860(1198861) 119891119860(1198862) 119891

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

= 1 minus 119891⟨119860⟩ (119909)

(16)

Now we will show that ⟨119860⟩ is a vague filter of 119871(VF1) Suppose that 119909 119910 isin 119871 is arbitrary For every 120576 gt 0

we can take 1198861 119886

119899 1198871 119887

119898isin 119871 such that119909 ge 119886

1lowastsdot sdot sdotlowast119886

119899

119910 ge 1198871lowast sdot sdot sdot lowast 119887

119898 and

119905⟨119860⟩ (119909) minus 120576 lt min 119905

119860(1198861) 119905119860(1198862) 119905

119860(119886119899)

119905⟨119860⟩

(119910) minus 120576 lt min 119905119860(1198871) 119905119860(1198872) 119905

119860(119887119898)

(17)

By Proposition 2 part (4) we have 119909 lowast 119910 ge 1198861lowast sdot sdot sdot lowast 119886

119899lowast 1198871lowast

sdot sdot sdot lowast 119887119898and then

119905⟨119860⟩

(119909 lowast 119910)

ge min 119905119860(1198861) 119905

119860(119886119899) 119905119860(1198871) 119905

119860(119887119898)

= min min 119905119860(1198861) 119905

119860(119886119899)

min 119905119860(1198871) 119905

119860(119887119898)

gt min (119905⟨119860⟩ (119909) minus 120576) (119905

⟨119860⟩(119910) minus 120576)

= min 119905⟨119860⟩ (119909) 119905⟨119860⟩ (119910) minus 120576

(18)

Since 120576 gt 0 is arbitrary we obtain that 119905⟨119860⟩

(119909 lowast 119910) ge

min119905⟨119860⟩

(119909) 119905⟨119860⟩

(119910)Similarly we can show that 1 minus 119891

⟨119860⟩(119909 lowast 119910) ge min1 minus

119891⟨119860⟩

(119909) 1 minus119891⟨119860⟩

(119910) Hence119881119860(119909lowast119910) ge imin119881

119860(119909) 119881119860(119910)

for all 119909 119910 isin 119871(VF2) Suppose that 119909 le 119910where 119909 119910 isin 119871 Let 119886

1 119886

119899isin

119871 such that 1198861lowast sdot sdot sdot lowast 119886

119899le 119909 Then 119886

1lowast sdot sdot sdot lowast 119886

119899le 119910 And we

get that

119881119860(119910) ge imin 119881

119860(1198861) 119881119860(1198862) 119881

119860(119886119899) (19)

Since 1198861 119886

119899isin 119871 where 119909 ge 119886

1lowast sdot sdot sdot lowast 119886

119899is arbitrary then

119881⟨119860⟩

(119910) ge isup imin 119881119860(1198861) 119881119860(1198862) 119881

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

= 119881⟨119860⟩ (119909)

(20)

Therefore ⟨119860⟩ is a vague filter of 119871 by Theorem 16 It is clearthat119860 ⪯ ⟨119860⟩ Now let119862 be a vague filter of119871 such that119860 ⪯ 119862Then we have

119881⟨119860⟩ (119909)

= isup imin 119881119860(1198861) 119881

119860(119886119899) 119909 ge 119886

1lowast sdot sdot sdot lowast 119886

119899

le isup imin 119881119862(1198861) 119881

119862(119886119899) 119909 ge 119886

1lowast sdot sdot sdot lowast 119886

119899

= 119881119862 (119909)

(21)

for all 119909 isin 119871 Hence ⟨119860⟩ ⪯ 119862 and then ⟨119860⟩ is a vague filtergenerated by 119860

We denote the set of all vague filters of a residuated lattice119871 by VF(119871)

6 Journal of Discrete Mathematics

Corollary 28 (VF(119871) cap or) is a complete lattice

Proof Clearly if 119860119894119894isinΓ

is a family of vague filters of aresiduated lattice 119871 then the infimumof this family is⋂

119894isinΓ119860119894

and the supremum is ⋁119894isinΓ

119860119894= ⟨⋃119894isinΓ

119860119894⟩ It is easy to prove

that (IFF(119871) cap or) is a complete lattice

Let 119860 and 119861 be vague sets of a residuated lattice 119871 Then119860 lowast 119861 is defined as follows

119881(119860lowast119861) (119911) = isup imin 119881

119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910 (22)

for all 119911 isin 119871

Theorem 29 Let 119860 and 119861 be vague sets of a residuated lattice119871 Then 119860 lowast 119861 is a vague set of 119871

Proof Since 119860 and 119861 are vague sets of 119871 we have 119905119860(119909) le

1 minus 119891119860(119909) and 119905

119861(119910) le 1 minus 119891

119861(119910) for all 119909 119910 isin 119871 Thus

119905119860lowast119861 (119911) = sup min 119905

119860 (119909) 119905119860 (119910) 119911 ge 119909 lowast 119910

le sup min 1 minus 119891119860 (119909) 1 minus 119891

119860(119910) 119911 ge 119909 lowast 119910

le 1 minus 119891119860lowast119861 (119911)

(23)

Hence 119905119860lowast119861

(119911) + 119891119860lowast119861

(119911) le 1 and then 119860 lowast 119861 is a vague set of119871

Theorem30 Let119860 and119861 be vague filters of a residuated lattice119871 Then 119860 lowast 119861 is a vague filter of 119871

Proof ByTheorem 29119860lowast119861 is a vague set of 119871 We will provethat 119860 lowast 119861 is a vague filter of 119871

(VF2) Let 1199111

le 1199112and 119909 lowast 119910 le 119911

1 We get that

imin119881119860(119909) 119881119861(119910) le 119881

119860lowast119861(1199112) Since 119909 lowast 119910 le 119911

1is arbitrary

we have

119881119860lowast119861

(1199111) = isup imin 119881

119860 (119909) 119881119861 (119910) 1199111ge 119909 lowast 119910

le 119881119860lowast119861

(1199112)

(24)

(VF1) Let 119909 119910 isin 119871 Then

imin 119881119860lowast119861 (119909) 119881119860lowast119861 (119910)

= imin isup imin 119881119860 (119886) 119881119861 (119887) 119909 ge 119886 lowast 119887

isup imin 119881119860 (119888) 119881119861 (119889) 119910 ge 119888 lowast 119889

= isup isup imin 119881119860 (119886 lowast 119888) 119881119861 (119888 lowast 119889)

119909 ge 119886 lowast 119887 119910 ge 119888 lowast 119889

le isup imin 119881119860 (119886 lowast 119888) 119881119861 (119888 lowast 119889)

119909 lowast 119910 ge (119886 lowast 119887) lowast (119888 lowast 119889)

le isup imin 119881119860 (119890) 119881119861 (119891) 119909 lowast 119910 ge 119890 lowast 119891

= 119881119860lowast119861

(119909 lowast 119910)

(25)

By Definition 13 119860 lowast 119861 is a vague filter of 119871

Theorem31 Let119860 and119861 be vague filters of a residuated lattice119871 such that119881

119860(1) = 119881

119861(1) Then119860lowast119861 is the least upper bound

of 119860 and 119861

Proof By Definition 13 we have 119881119860(1) = 119881

119861(1) ge 119881

119860(119910) for

all 119910 isin 119871 Let 119911 isin 119871 Since 119911 = 1 lowast 119911 we get that

119881119860lowast119861 (119911) = isup imin 119881

119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910

ge imin 119881119860 (119911) 119881119861 (1) ge 119881

119860 (119911)

(26)

Hence 119860 ⪯ 119860 lowast 119861 Similarly we can show that 119861 ⪯ 119860 lowast 119861Suppose that 119862 is the vague filter of 119871 containing 119860 and 119861Then

119881119860lowast119861 (119911)

= isup imin 119881119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910

le isup imin 119881119862 (119909) 119881119862 (119910) 119911 ge 119909 lowast 119910

le isup imin 119881119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 = 120583

119862 (119911)

(27)

Hence 119860lowast 119861 ⪯ 119862 that is 119860lowast 119861 is the least upper bound of 119860and 119861

Lemma 32 Let119862 be a vague filter of a residuated lattice 119871 and119911 isin 119871 Then

(1) 119881119862(119911) = isupVC(x lowast y) z ge x lowast y for all 119909 119910 119911 isin 119871

(2) 119881119862(119909 lowast 119910) = iminVC(x)VC(y) for 119909 119910 isin 119871

Proof (1) Since 119909 lowast 119910 le 119911 then 119881119862(119909 lowast 119910) le 119881

119862(119911) by (VF2)

Hence 119881119862(119911) ge isup119881

119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 Also we have

1 lowast 119911 = 119911 We obtain that

119881119862 (119911) = 119881

119862 (1 lowast 119911) le isup 119881119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 (28)

(2) It follows from Proposition 2 part (2) andDefinition 13

Let VFD(119871) be the set of all vague filters119860 and119861 of 119871 suchthat 119881

119860(1) = 119881

119861(1)

Theorem 33 (119881119865119863(119871) cap lowast) is a distributive lattice

Journal of Discrete Mathematics 7

Proof Let 119860 119861 119862 isin VFD(119871) be such that 119860 ⪯ 119862 We willshow that (119860 lowast 119861) cap 119862 = (119860 lowast 119862) cap (119861 lowast 119862) By Lemma 32

119881(119860lowast119861)cap119862 (119911)

= imin (119881119860lowast119861

) (119911) 119881119862 (119911)

= imin isup imin 119881119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910 119881

119862 (119911)

= isup imin 119881119860 (119909) 119881119861 (119910) 119881

119862 (119911) 119911 ge 119909 lowast 119910

= isup imin 119881119860 (119909) 119881119861 (119910) 119881

119862(119909 lowast 119910) 119911 ge 119909 lowast 119910

= isup imin 119881119860 (119909) 119881119861 (119910) 119881

119862 (119909) 119881119862 (119910) 119911 ge 119909 lowast 119910

= isup imin 119881119860 (119909) 119881119862 (119909)

imin 119881119861(119910) 119881

119862(119910) 119911 ge 119909 lowast 119910

= isup imin 119881119860cap119862 (119909) 119881119861cap119862 (119910) 119911 ge 119909 lowast 119910

= 119881(119860cap119862)lowast(119861cap119862) (119911)

(29)

5 Subpositive Implicative Vague Filters

Definition 34 Let 119860 be a vague subset of a residuated lattice119871 119860 is called a subpositive implicative vague filter of 119871 if

(VF6) 119881119860(119909) le 119881

119860(1) for all 119909 isin 119871

(SVF1) imin119881119860(((119909 rarr 119910) lowast 119911) 999492999484 ((119910 999492999484 119909) rarr

119909)) 119881119860(119911) le 119881

119860((119909 rarr 119910) 999492999484 119910) for any

119909 119910 119911 isin 119871(SVF2) imin119881

119860((119911 lowast (119909 999492999484 119910)) rarr ((119910 rarr 119909) 999492999484

119909)) 119881119860(119911) le 119881

119860((119909 999492999484 119910) rarr 119910) for any

119909 119910 119911 isin 119871

Proposition 35 Let 119860 be a vague filter of a residuated lattice119871 Then 119860 is a subpositive implicative vague filter of L if andonly if it satisfies

(SVF3) 119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) le

119881119860((119909 rarr 119910) 999492999484 119910) for any 119909 119910 isin 119871

(SVF4) 119881119860((119909 999492999484 119910) rarr ((119910 rarr 119909) 999492999484 119909)) le

119881119860((119909 999492999484 119910) rarr 119910) for any 119909 119910 isin 119871

Proposition 36 Let119860 be a subpositive implicative vague filterof a residuated lattice 119871 Then 119860 is a vague filter of 119871

Proof We have imin119881119860(119909 999492999484 119910) 119881

119860(119909) = imin119881

119860((119910 rarr

119910) lowast 119909) 999492999484 ((119910 999492999484 119910) rarr 119910) 119881119860(119909) le 119881

119860((119910 rarr 119910) 999492999484

119910) = 119881119860(119910) By (VS1) andTheorem 17 119860 is a vague filter of a

residuated lattice 119871

The relation between subpositive implicative vague filterand its level subset is as follows

Theorem 37 Let 119860 be a vague set of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and only if

for all120572 120573 isin [0 1] the set119860(120572120573)

is either empty or a subpositiveimplicative filter of 119871 where 120572 le 120573

Proof Suppose that119860 is a subpositive implicative filter vagueset of 119871 and 120572 120573 isin [0 1] Let 119860

(120572120573)= 0 Since 119860 is a vague

filter then119860(120572120573)

is a filter of 119871 byTheorem 21 Now let (119909 rarr

119910) 999492999484 ((119910 999492999484 119909) rarr 119909) isin 119860(120572120573)

Then 119881119860((119909 rarr 119910) 999492999484

((119910 999492999484 119909) rarr 119909)) ge [120572 120573] By (SVF3) 119881119860((119909 rarr 119910) 999492999484

119910) ge [120572 120573] that is (119909 rarr 119910) 999492999484 119910 isin 119860(120572120573)

Similarly we canshow that119860

(120572120573)satisfies (SVF4) Hence119860

(120572120573)is a subpositive

implicative filter of 119871Conversely let 119905

119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) =

120572 and 1 minus 119891119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) = 120573 Thus

119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) ge [120572 120573] We get that

(119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909) isin 119860(120572120573)

So 119860(120572120573)

= 0Thus (119909 rarr 119910) 999492999484 119910 isin 119860

(120572120573) that is 119881

119860((119909 rarr 119910) 999492999484 119910) ge

[120572 120573] = 119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) Similarly we can

prove that 119860 satisfies (SVF4)

In the following theorems we obtain some characteriza-tions of subpositive implicative vague filter

Theorem 38 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF5) iminVA((y rarr z) 999492999484 (x rarr y))VA(x) le

VA(y) for any 119909 119910 119911 isin 119871(SVF6) iminVA((y 999492999484 z) rarr (x 999492999484 y))VA(x) le

VA(y) for any 119909 119910 119911 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871 ByProposition 2 part (3) we have imin119881

119860((119910 rarr 119911) 999492999484 (119909 rarr

119910)) 119881119860(119909) = imin119881

119860(119909 rarr ((119910 rarr 119911) 999492999484 119910)) 119881

119860(119909) le

119881119860((119910 rarr 119911) 999492999484 119910) By Proposition 2 part (7) (119910 rarr 119911) 999492999484

119910 le 119911 999492999484 119910 le (119910 rarr 119911) 999492999484 ((119911 999492999484 119910) rarr 119910) We get that119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119911 999492999484 119910) le 119881

119860((119910 rarr 119911) 999492999484 ((119911 999492999484

119910) rarr 119910)) le 119881119860((119911 999492999484 119910) rarr 119910) Hence imin119881

119860((119910 rarr

119911) 999492999484 119910) 119881119860(119911 999492999484 119910) le imin119881

119860((119911 999492999484 119910) rarr 119910) 119881

119860(119911 999492999484

119910) le 119881119860(119910) Thus it satisfies (SVF5) Similarly we can

prove (SVF6) Conversely let 119860 satisfy (SVF5) and (SVF6)119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) = imin119881

119860([((119909 rarr 119910) 999492999484

119910) rarr 1] 999492999484 [(119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909) rarr ((119909 rarr

119910) 999492999484 119910)]) 119881119860(1) le 119881

119860((119909 rarr 119910) 999492999484 119910) Similarly we

can prove that (SVF5) holds Hence 119860 is a vague subpositiveimplicative filter

Theorem 39 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF7) 119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) le

119881119860((119910 999492999484 119909) rarr 119909) for any 119909 119910 isin 119871

(SVF8) 119881119860((119909 999492999484 119910) rarr ((119910 999492999484 119909) rarr 119909)) le

119881119860((119910 rarr 119909) 999492999484 119909) for any 119909 119910 isin 119871

Proof Let (SVF7) and (SVF8) hold We will show that 119860

satisfies (SVF5) and (SVF6) We have imin119881119860((119910 rarr 119911) 999492999484

(119909 rarr 119910)) 119881119860(119909) = imin119881

119860(119909 rarr ((119910 rarr 119911) 999492999484

119910)) 119881119860(119909) le 119881

119860((119910 rarr 119911) 999492999484 119910) le 119881

119860((119910 rarr 119911) 999492999484

8 Journal of Discrete Mathematics

((119911 rarr 119910) 999492999484 119910)) le 119881119860((119911 999492999484 119910) rarr 119910)) Also we have

119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119911 999492999484 119910) Since 119860 is a vague

filter 119881119860((119910 rarr 119911) 999492999484 119910) le imin119881

119860(119911 999492999484 119910) 119881

119860((119911 999492999484

119910) rarr 119910) le 119881119860(119910) Hence 119860 satisfies (SVF5) Analogously

we can prove that119860 satisfies (SVF6) Hence119860 is a subpositiveimplicative vague filter of 119871 byTheorem 38 Conversely let 119860be a subpositive implicative vague filter of 119871 We have

[((119910 999492999484 119909) 997888rarr 119909) 997888rarr 1]

999492999484 [((119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909))

997888rarr ((119910 999492999484 119909) 997888rarr 119909) ]

= [(119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)]

997888rarr [((119910 997888rarr 119909) 999492999484 119909) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)]

= (119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)

(30)

We get that 119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) =

imin119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909))119881

119860([((119910 999492999484 119909) rarr

119909) rarr 1] 999492999484 [((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) rarr ((119910 999492999484

119909) rarr 119909)]) le 119881119860((119910 999492999484 119909) rarr 119909) by (SVF5) Similarly we

can prove that (SVF8) holds

Theorem 40 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF9) 119881119860((119909 rarr 119910) 999492999484 119909) le 119881

119860(119909) for any 119909 119910 isin 119871

(SVF10) 119881119860((119909 999492999484 119910) rarr 119909) le 119881

119860(119909) for any 119909 119910 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871Then 119860 satisfies (SVF5) and (SVF6)

119881119860((119909 997888rarr 119910) 999492999484 119909)

= imin 119881119860(1 997888rarr ((119909 997888rarr 119910) 999492999484 119909)) 119881

119860 (1)

= imin 119881119860((119909 997888rarr 119910) 999492999484 (1 997888rarr 119909)) 119881119860 (1)

le 119881119860 (119909)

(31)

Similarly we can prove that (SVF10) holds Conversely let119860 satisfy (SVF9) and (SVF10) We will show that 119860 satisfies(SVF5) and (SVF6) imin119881

119860((119910 rarr 119911) 999492999484 (119909 rarr

119910)) 119881119860(119909) = imin119881

119860((119909 rarr (119910 rarr 119911) 999492999484 119910)) 119881

119860(119909) le

119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119910) Hence 119860 satisfies (SVF5)

Similarly we can prove (SVF7) holds Therefore 119860 is asubpositive implicative vague filter of L byTheorem 38

Theorem 41 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF11) 119881119860(not119909 999492999484 119909) le 119881

119860(119909) for any 119909 isin 119871

(SVF12) 119881119860(sim 119909 rarr 119909) le 119881

119860(119909) for any 119909 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871Then (SVF11) and (SVF12) follow by (SVF9) and (SVF10)Conversely since 0 le 119910 then 119909 rarr 0 le 119909 rarr 119910 Hence

(119909 rarr 119910) 999492999484 119909 le not119909 999492999484 119909 by Proposition 2 part (6)Therefore 119881

119860((119909 rarr 119910) 999492999484 119909) le 119881

119860(not119909 999492999484 119909) le 119881

119860(119909)

Similarly we can prove that (SVF10) holds Hence 119860 is asubpositive implicative vague filter of 119871 byTheorem 40

The extension theorem of subpositive implicative vaguefilters is obtained from the following theorem

Theorem 42 Let 119860 and 119861 be two vague filters of a residuatedlattice 119871 which satisfies 119860 le 119861 and 119881

119860(1) = 119881

119861(1) If 119860 is a

subpositive implicative vague filter of L so 119861 is

Proof Suppose that 119906 = (119909 rarr 119910) 999492999484 119909 Then

1 = 119906 997888rarr 119906 = 119906 997888rarr ((119909 997888rarr 119910) 999492999484 119909)

= (119909 997888rarr 119910) 999492999484 (119906 997888rarr 119909) = 1

(32)

We have (119909 rarr 119910) 999492999484 (119906 rarr 119909) le ((119906 rarr 119909) rarr 119910) 999492999484

(119906 rarr 119909) Therefore 119881119860(((119906 rarr 119909) rarr 119910) 999492999484 (119906 rarr 119909)) =

119881119860(1) By (SVF10) 119881

119860(1) = 119881

119860(((119906 rarr 119909) rarr 119910) 999492999484 (119906 rarr

119909)) le 119881119860(119906 rarr 119909) Hence 119881

119860(119906 rarr 119909) = 119881

119860(1) = 119881

119861(1)

Since 119860 le 119861 then 119881119861(119906 rarr 119909) ge 119881

119860(119906 rarr 119909) = 119881

119861(1)

We get that 119881119861((119909 rarr 119910) 999492999484 119909) = 119881

119861(119906) = imin119881

119861(119906 rarr

119909) 119881119861(119906) le 119881

119861(119909) Similarly we can show that 119866 satisfies

(SVF11) Hence 119866 is a vague subpositive implicative filter of119871 by Theorem 40

Definition 43 Let 119860 be a vague filter of a residuated lattice 119871119860 is called a vague Boolean filter of 119871 if

(BVF1) 119881119860(119909 or not119909) = 119881

119860(1) for all 119909 isin 119871

(BVF2) 119881119860(119909or sim 119909) = 119881

119860(1) for all 119909 isin 119871

Similar to Theorem 40 we have the following theoremwith respect to Boolean vague filters

Theorem 44 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a Boolean vague filter of 119871 if and only if for all120572 120573 isin [0 1] the set 119860

(120572120573)is either empty or a Boolean filter of

119871 where 120572 le 120573

Theorem 45 Let 119860 be a vague filter of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and if onlyif 119860 is a Boolean vague filter

Proof Let119860 be a subpositive implicative vague filterWe have

119881119860 (1) = 119881

119860 (not (119909 or not119909) 999492999484 (119909 or not119909))

= 119881119860 ((not119909 and notnot119909) 999492999484 (119909 or not119909)) le 119881

119860 (119909 or not119909)

(33)

Hence 119881119860(119909 or not119909) = 119881

119860(1) Similarly we can prove that 119860

satisfies (BVF2) Hence 119860 is a Boolean vague filterConversely let119881

119860(119909ornot119909) = 119881

119860(1)We have119881

119860(119909ornot119909) =

119881119860((119909 999492999484 119909) or (not119909 999492999484 119909))119881

119860((119909 or not119909) 999492999484 119909) = imin119881

119860((119909 or

not119909) 999492999484 119909) 119881119860(1) = imin119881

119860((119909 or not119909) 999492999484 119909) 119881

119860(119909 or not119909) le

119881119860(119909) Similarly we can show that119860 satisfies (SVF12) Hence

119860 is a subpositive implicative vague filter byTheorem 41

Journal of Discrete Mathematics 9

6 Conclusion

In this paper we introduced the notions of vague filterssubpositive implicative vague filters and Boolean vaguefilters of a residuated lattice and investigated some relatedproperties We studied the relationship between filters andvague filters of a residuated lattice and showed that vaguefilters of a residuated lattice is a generalization of fuzzy filtersWe proved that the set of all vague filters of a residuated latticeforms a complete lattice and obtained some characterizationsof subpositive implicative vague filterThe extension theoremof subpositive implicative vague filters was obtained Finallyit was proved that subpositive implicative vague filters areequivalent to Boolean vague filters

Conflict of Interests

There is no conflict of interests regarding the publication ofthis paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993

[3] Y B Jun and C H Park ldquoIdeals of pseudo MV-algebras basedon vague set theoryrdquo Iranian Journal of Fuzzy Systems vol 6 no7 pp 31ndash47 2009

[4] A Borumand Saeid ldquoVague BCKBCI-algebrasrdquo OpusculaMathematica vol 29 no 2 pp 177ndash186 2009

[5] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939

[6] T S Blyth and M F Janowitz Residuation Theory PergamonPress 1972

[7] T Kowalski and H Ono Residuated Lattices An AlgebraicGlimpse at Logic without Contraction Japan Advanced Instituteof Science and Technology March 2001

[8] D PiciuAlgebras of Fuzzy Logic Editura Universitaria CraiovaCraiova Romania 2007

[9] E TurunenMathematics Behind Fuzzy Logic Physica 1999[10] P Hajek Metamathematics of Fuzzy Logic vol 4 of Trends

in LogicmdashStudia Logica Library Kluwer Academic DordrechtThe Netherlands 1998

[11] Sh Ghorbani ldquoSub positive implicative filters of residuatedlatticesrdquo World Applied Science Journal vol 12 no 5 pp 586ndash590 2011

[12] S Ghorbani and A Hasankhani ldquoFuzzy convex subalgebrasof commutative residuated latticesrdquo Iranian Journal of FuzzySystems vol 7 no 2 pp 41ndash54 159 2010

[13] R Biswas ldquoVague groupsrdquo International Journal of Computa-tional Cognition vol 4 pp 20ndash23 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Discrete Mathematics

vague filters and Boolean vague filters of a residuated latticeare introduced and some related properties are obtained

2 Preliminaries

We recall some definitions and theorems which will beneeded in this paper

Definition 1 (see [6]) A residuated lattice is an algebraicstructure (119871 and or lowast rarr 999492999484 0 1) such that

(1) (119871 and or 0 1) is a bounded lattice with the least ele-ment 0 and the greatest element 1

(2) (119871 lowast 1) is amonoid that islowast is associative and119909lowast1 =

1 lowast 119909 = 119909

(3) 119909 lowast 119910 le 119911 if and only if 119909 le 119910 rarr 119911 if and only if119910 le 119909 999492999484 119911 for all 119909 119910 119911 isin 119871

In the rest of this paper we denote the residuated lattice(119871 and or lowast rarr 999492999484 0 1) by 119871 not119909 = 119909 rarr 0 and sim 119909 = 119909 999492999484 0

Proposition 2 (see [6 9]) Let 119871 be a residuated lattice Thenwe have the following properties for all 119909 119910 119911 isin 119871

(1) 119909 rarr 119909 = 119909 999492999484 119909 = 1

(2) 1 rarr 119909 = 1 999492999484 119909 = 119909

(3) 119909 rarr (119910 999492999484 119911) = 119910 999492999484 (119909 rarr 119911)

(4) 119909 le 119910 if and only if 119909 rarr 119910 = 1 if and only if 119909 999492999484 119910 =

1

(5) if 119909 le 119910 then 119909 lowast 119911 le 119910 lowast 119911 119911 lowast 119909 le 119911 lowast 119910

(6) if 119909 le 119910 then 119911 rarr 119909 le 119911 rarr 119910 119911 999492999484 119909 le 119911 999492999484 119910

(7) if 119909 le 119910 then 119910 rarr 119911 le 119909 rarr 119911 119910 999492999484 119911 le 119909 999492999484 119911

(8) 119909 le (119909 999492999484 119910) rarr 119910 119909 le (119909 rarr 119910) 999492999484 119910

(9) 119910 rarr 119911 le (119909 rarr 119910) 999492999484 (119909 rarr 119911) 119910 999492999484 119911 le (119909 999492999484

119910) 999492999484 (119909 999492999484 119911)

(10) 119909 rarr (119910 999492999484 (119909 lowast 119910)) = 1 119910 999492999484 (119909 rarr (119909 lowast 119910)) = 1

Definition 3 (see [6 9]) Let 119865 be a nonempty subset of aresiduated lattice 119871 119865 is called a filter if

(F1) 119909 119910 isin 119865 imply 119909 lowast 119910 isin 119865

(F2) 119909 le 119910 and 119909 isin 119865 imply 119910 isin 119865

Theorem 4 (see [6 9]) A nonempty subset 119865 of a residuatedlattice 119871 is a filter if and only if

(F3) 1 isin 119865 and 119909 119909 rarr 119910 isin 119865 imply 119910 isin 119865 or

(F31015840) 1 isin 119865 and 119909 119909 999492999484 119910 isin 119865 imply 119910 isin 119865

Definition 5 (see [11]) Let 119865 be a nonempty subset of aresiduated lattice 119871 119865 is called a subpositive implicative filterof 119871 if

(SPIF1) 1 isin 119865(SPIF2) ((119909 rarr 119910) lowast 119911) 999492999484 ((119910 999492999484 119909) rarr 119909) isin 119865

and 119911 isin 119865 imply (119909 rarr 119910) 999492999484 119910 isin 119865 for any119909 119910 119911 isin 119871

(SPIF3) (119911 lowast (119909 999492999484 119910)) rarr ((119910 rarr 119909) 999492999484 119909) isin 119865

and 119911 isin 119865 imply (119909 999492999484 119910) rarr 119910 isin 119865 for any119909 119910 119911 isin 119871

Proposition 6 (see [11]) Let119865 be a filter of119871119865 is a subpositiveimplicative filter if and only if

(SPIF4) (119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909) isin 119865 imply(119909 rarr 119910) 999492999484 119910 isin 119865 for any 119909 119910 isin 119871

(SPIF5) (119909 999492999484 119910) rarr ((119910 rarr 119909) 999492999484 119909) isin 119865 imply(119909 999492999484 119910) rarr 119910 isin 119865 for any 119909 119910 isin 119871

Definition 7 (see [1]) Let119883 be a nonempty set A fuzzy set in119883 is a mapping 120583 119883 rarr [0 1]

Proposition 8 (see [12]) Let 119871 be a residuated lattice A fuzzyset 120583 is called a fuzzy filter in 119871 if it satisfies

(fF1) 119909 le 119910 imply 120583(119909) le 120583(119910)

( fF2) min120583(119909) 120583(119910) le 120583(119909 lowast 119910)

for all 119909 119910 isin 119871

Definition 9 (see [13]) A vague set 119860 in the universe ofdiscourse 119883 is characterized by two membership functionsgiven by

(i) a true membership function 119905119860

119883 rarr [0 1]

(ii) a false membership function 119891119860

119883 rarr [0 1]

where 119905119860(119909) is a lower bound on the grade of membership of

119883 derived from the ldquoevidence for 119909rdquo 119891119860(119909) is a lower bound

on the negation of 119909 derived from the ldquoevidence against 119909rdquoand 119905119860(119909) + 119891

119860(119909) le 1

Thus the grade of membership of 119909 in the vague set 119860

is bounded by a subinterval [119905119860(119909) 1 minus 119891

119860(119909)] of [0 1] This

indicates that if the actual grade of membership of 119909 is 120583(119909)then 119905

119860(119909) le 120583(119909) le 1 minus 119891

119860(119909) The vague set 119860 is written

as 119860 = ⟨119909 [119905119860(119909) 1 minus 119891

119860(119909)]⟩ 119909 isin 119883 where the interval

[119905119860(119909) 1 minus 119891

119860(119909)] is called the vague value of 119909 in 119860 denoted

by 119881119860(119909)

Definition 10 (see [13]) A vague set 119860 of a set 119883 is called

(1) the zero vague set of 119883 if 119905119860(119909) = 0 and 119891

119860(119909) = 1 for

all 119909 isin 119883

(2) the unit vague set of 119883 if 119905119860(119909) = 1 and 119891

119860(119909) = 0 for

all 119909 isin 119883

(3) the 120572-vague set of 119883 if 119905119860(119909) = 120572 and 119891

119860(119909) = 1 minus 120572

for all 119909 isin 119883 where 120572 isin (0 1)

Let 119868[0 1] denote the family of all closed subintervals of[0 1] Nowwedefine the refinedminimum (briefly imin) and

Journal of Discrete Mathematics 3

an order le on elements 1198681= [1198861 1198871] and 119868

2= [1198862 1198872] of 119868[0 1]

asimin (119868

1 1198682) = [min 119886

1 1198862 min 119887

1 1198872]

1198681le 1198682

iff 1198861le 1198862 1198871le 1198872

(1)

Similarly we can define ge = and imax Then the concept ofimin and imax could be extended to define iinf and isup ofinfinite number of elements of 119868[0 1] It is a known fact that119871 = 119868[0 1] iinf isup le is a lattice with universal boundszero vague set and unit vague set For 120572 120573 isin [0 1] we nowdefine (120572 120573)-cut and 120572-cut of a vague set

Definition 11 (see [13]) Let 119860 be a vague set of a universe 119883

with the true-membership function 119905119860and false-membership

function119891119860The (120572 120573)-cut of the vague set119860 is a crisp subset

119860(120572120573)

of the set 119883 given by

119860(120572120573)

= 119909 isin 119883 119881119860 (119909) ge [120572 120573]

where 120572 le 120573

(2)

Clearly 119860(00)

= 119883 The (120572 120573)-cuts are also called vague-cutsof the vague set 119860

Definition 12 (see [13]) The 120572-cut of the vague set119860 is a crispsubset 119860

120572of the set119883 given by 119860

120572= 119860(120572120572)

Equivalently wecan define the 120572-cut as 119860

120572= 119909 isin 119883 119905

119860(119909) ge 120572

3 Vague Filters of Residuated Lattices

In this section we define the notion of vague filters ofresiduated lattices and obtain some related results

Definition 13 A vague set 119860 of a residuated lattice 119871 is calleda vague filter of 119871 if the following conditions hold

(VF1) if 119909 le 119910 then 119881119860(119909) le 119881

119860(119910) for all 119909 119910 isin 119871

(VF2) imin119881119860(119909) 119881119860(119910) le 119881

119860(119909lowast119910) for all 119909 119910 isin

119871

That is(1) 119909 le 119910 implies 119905

119860(119909) le 119905

119860(119910) and 1minus119891

119860(119909) le 1minus119891

119860(119910)

for all 119909 119910 isin 119871(2) min119905

119860(119909) 119905119860(119910) le 119905

119860(119909 lowast 119910) and min1 minus 119891

119860(119909) 1 minus

119891119860(119910) le 1 minus 119891

119860(119909 lowast 119910) for all 119909 119910 isin 119871

In the following proposition we will show that the vaguefilters of a residuated lattice exist

Proposition 14 Unit vague set and120572-vague set of a residuatedlattice 119871 are vague filters of 119871

Proof Let 119860 be a 120572-vague set of 119871 It is clear that 119860 satisfies(VF1) For 119909 119910 isin 119871 we have

119905119860(119909 lowast 119910) = 120572 = min 120572 120572 = min 119905

119860 (119909) 119905119860 (119910)

1 minus 119891119860(119909 lowast 119910)

= 120572 = min 120572 120572 = min 1 minus 119891119860 (119909) 1 minus 119891

119860(119910)

(3)

Thus imin119881119860(119909) 119881119860(119910) le 119881

119860(119909 lowast 119910) for all 119909 119910 isin 119871 Hence

it is a vague filter of 119871 The proof of the other cases is similar

Lemma 15 Let 119860 be a vague set of a residuated lattice 119871 suchthat it satisfies (VF1) Then the following are equivalent

(VF4) iminVA(x)VA(x rarr y) le VA(y)(VF5) iminVA(x)VA(x 999492999484 y) le VA(y)

for all 119909 119910 isin 119871

Proof Suppose that 119860 satisfies (VF4) We have 119909 le (119909 999492999484

119910) rarr 119910 By (VF1) 119881119860(119909) le 119881

119860((119909 999492999484 119910) rarr 119910) We get that

imin 119881119860 (119909) 119881119860 (119909 999492999484 119910)

le imin 119881119860((119909 999492999484 119910) 997888rarr 119910) 119881

119860(119909 999492999484 119910)

le 119881119860(119910)

(4)

Conversely if119860 satisfies (VF5) then the inequality 119909 le (119909 rarr

119910) 999492999484 119910 analogously entails (VF4)

Theorem 16 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter if and only if it satisfies (VF4) and

(VF6) 119881119860(119909) le 119881

119860(1) for all 119909 isin 119871

Proof Let 119860 be a vague filter of 119871 Since 119909 le 1 then 119881119860(119909) le

119881119860(1) for all 119909 isin 119871 by (VF1) We have 119909 lowast (119909 rarr 119910) le 119910 By

(VF1) and (VF2) we get

imin 119881119860 (119909) 119881119860 (119909 997888rarr 119910)

le 119881119860(119909 lowast (119909 997888rarr 119910)) le 119881

119860(119910)

(5)

Hence (VF4) holdConversely let 119860 satisfy (VF4) and (VF6) Since 119909 le 119910

then 119909 rarr 119910 = 1

119881119860(119910) ge imin 119881

119860(119909 997888rarr 119910) 119881

119860 (119909)

= imin 119881119860 (1) 119881119860 (119909) = 119881

119860 (119909)

(6)

Hence 119881119860

is order-preserving and (VF1) holds ByProposition 2 part (10) 119909 rarr (119910 999492999484 (119909 lowast 119910)) = 1 By(VF4) we have

119881119860(119910 999492999484 (119909 lowast 119910))

ge imin 119881119860 (119909) 119881119860 (119909 997888rarr (119910 999492999484 (119909 lowast 119910))) = 119881

119860 (119909)

(7)

By Lemma 15

119881119860(119909 lowast 119910) ge imin 119881

119860(119910 999492999484 (119909 lowast 119910)) 119881

119860(119910)

ge imin 119881119860 (119909) 119881119860 (119910)

(8)

Hence (VF2) holds

Theorem 17 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter if and only if it satisfies (VF5) and(VF6)

4 Journal of Discrete Mathematics

Definition 18 A vague set 119860 of a residuated lattice 119871 is calleda vague lattice filter of 119871 if it satisfies (VF1) and

(VLF2) imin119881119860(119909) 119881119860(119910) le 119881

119860(119909 and 119910) for all

119909 119910 isin 119871

Proposition 19 Let 119860 be a vague filter of a residuated lattice119871 then 119860 is a vague lattice filter

In the following example we will show that the converseof Proposition 19 may not be true in general

Example 20 Let 0 119886 119887 119888 1 such that 0 lt 119886 lt 119887 119888 lt 1where 119887 and 119888 are incomparable Define the operations lowast rarr and 999492999484 as follows

lowast 0 119886 119887 119888 1

0 0 0 0 0 0

119886 0 0 0 119886 119886

119887 0 119886 119887 119886 119887

119888 0 0 0 119888 119888

1 0 119886 119887 119888 1

997888rarr 0 119886 119887 119888 1

0 1 1 1 1 1

119886 119888 1 1 1 1

119887 119888 119888 1 119888 1

119888 0 119887 119887 1 1

1 0 119886 119887 119888 1

999492999484 0 119886 119887 119888 1

0 1 1 1 1 1

119886 119887 1 1 1 1

119887 0 119888 1 119888 1

119888 119887 119887 119887 1 1

1 0 119886 119887 119888 1

(9)

Then 119871 is a residuated lattice Let 119860 be the vague set definedas follows

119860 = ⟨0 [01 02]⟩ ⟨119886 [03 04]⟩ ⟨119887 [03 04]⟩

⟨119888 [03 08]⟩ ⟨0 [05 09]⟩

(10)

It is routine to verify that119860 is a vague lattice filter but it is nota vague filter of 119871

Let 119861 be the vague set defined as follows

119861 = ⟨0 [02 05]⟩ ⟨119886 [02 05]⟩ ⟨119887 [02 05]⟩

⟨119888 [04 07]⟩ ⟨0 [07 07]⟩

(11)

Then 119861 is a vague filter of 119871

In the following theorems we will study the relationshipbetween filters and vague filters of a residuated lattice

Theorem 21 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter of 119871 if and only if for all 120572 120573 isin [0 1]the set 119860

(120572120573)is either empty or a filter of 119871 where 120572 le 120573

Proof Suppose that 119860 is a vague filter of 119871 and 120572 120573 isin [0 1]Let 119860

(120572120573)= 0 We will show that 119860

(120572120573)is a filter of 119871

(1) Suppose that 119909 le 119910 and 119909 isin 119860(120572120573)

By (VF1)119881119860(119910) ge

119881119860(119909) ge [120572 120573] Therefor 119910 isin 119860

(120572120573)

(2) Suppose that 119909 119910 isin 119860(120572120573)

Then 119881119860(119909) 119881119860(119910) ge

[120572 120573] By (VF2) we have 119881119860(119909 lowast 119910) ge

imin119881119860(119909) 119881119860(119910) ge [120572 120573] Hence 119909 lowast 119910 isin 119860

(120572120573)

and then 119860(120572120573)

is a filter of 119871

Conversely suppose that for all 120572 120573 isin [0 1] the sets 119860(120572120573)

is either empty or a filter of 119871 Let 119905119860(119909) = 120572

1 119905119860(119910) = 120572

2

1 minus 119891119860(119909) = 120573

1 and 1 minus 119891

119860(119910) = 120573

2 Put 120572 = min120572

1 1205722

and 120573 = min1 minus 1205731 1 minus 120573

2 Then 119905

119860(119909) 119905119860(119910) ge 120572 and 1 minus

119891119860(119909) 1 minus 119891

119860(119910) ge 120573 Hence 119881

119860(119910) 119881119860(119909) ge [120572 120573] that is

119909 119910 isin 119860(120572120573)

So 119860(120572120573)

= 0 and by assumption 119860(120572120573)

is afilter of 119871 We obtain that 119909 lowast 119910 isin 119860

(120572120573) that is

119881119860(119909 lowast 119910) ge [120572 120573]

= [min 1205721 1205722 min 1 minus 120573

1 1 minus 120573

2]

= imin [119905119860 (119909) 1 minus 119891

119860 (119909)] [119905119860 (119910) 1 minus 119891119860(119910)]

= imin 119881119860 (119909) 119881119860 (119910)

(12)

Hence (VF2) holdsSuppose that 119909 le 119910 119905

119860(119909) = 120572 and 1 minus 119891

119860(119909) = 120573 Hence

119909 isin 119860(120572120573)

By assumption 119860(120572120573)

is a filter We get that 119910 isin

119860(120572120573)

that is 119881119860(119910) ge [120572 120573] = [119905

119860(119909) 1 minus 119891

119860(119909)] = 119881

119860(119909)

Hence 119860 is a vague filter of 119871

The filters like119860(120572120573)

are also called vague-cuts filters of 119871

Corollary 22 Let 119860 be a vague filter of a residuated lattice 119871Then for all 120572 isin [0 1] the set 119860

120572is either empty or a filter of

119871

Corollary 23 Any filter 119865 of a residuated lattice 119871 is a vague-cut filter of some vague filter of 119871

Proof For all 119909 isin 119871 define

1 minus 119891119860 (119909) = 119905

119860 (119909) = 1 if 119909 isin 119865

120572 otherwise(13)

Hence we have 119881119860(119909) = [1 1] if 119909 isin 119865 and 119881

119860(119909) = [120572 120572]

otherwise where 120572 isin (0 1) It is clear that 119865 = 119860(11)

Let119909 119910 isin 119871 If 119909 119910 isin 119865 then 119909 lowast 119910 isin 119865 We have 119881

119860(119909 lowast 119910) =

[1 1] = imin119881119860(119909) 119881119860(119910) If one of 119909 or 119910 does not belong

to 119865 then one of119881119860(119909) and119881

119860(119910) is equal to [120572 120572]Therefore

119881119860(119909lowast119910) ge [120572 120572] = imin119881

119860(119909) 119881119860(119910) Suppose that 119909 le 119910

If 119909 isin 119865 then 119910 isin 119865 Hence 119881119860(119909) = 119881

119860(119910) If 119909 notin 119865 then

119881119860(119909) = [120572 120572] We have 119881

119860(119909) le 119881

119860(119910) Hence 119860 is a vague

filter of 119871

In the following proposition we will show that vaguefilters of a residuated lattice are a generalization of fuzzyfilters

Proposition 24 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter of 119871 if and only if the fuzzy sets 119905

119860and

1 minus 119891119860are fuzzy filters in 119871

Journal of Discrete Mathematics 5

4 Lattice of Vague Filters

Let 119860 and 119861 be vague sets of a residuated lattice 119871 If 119881119860(119909) le

119881119861(119909) for all 119909 isin 119871 then we say 119861 containing119860 and denote it

by 119860 ⪯ 119861The unite vague set of a residuated lattice 119871 is a vague

filter of 119871 containing any vague filter of 119871 For a vague setwe can define the intersection of two vague sets 119860 and 119861 of aresiduated lattice 119871 by 119881

119860cap119861(119909) = imin119881

119860(119909) 119881119861(119909) for all

119909 isin 119871 It is easy to prove that the intersection of any family ofvague filters of 119871 is a vague filter of 119871 Hence we can define avague filter generated by a vague set as follows

Definition 25 Let 119860 be a vague set of a residuated lattice 119871A vague filter 119861 is called a vague filter generated by 119860 if itsatisfies

(i) 119860 ⪯ 119861(ii) 119860 ⪯ 119862 that implies 119861 ⪯ 119862 for all vague filters 119862 of 119871

The vague filter generated by 119860 is denoted by ⟨119860⟩

Example 26 Consider the vague lattice filter119860 in Example 20which is not a vague filter of 119871Then the vague filter generatedby 119860 is

⟨119860⟩ = ⟨0 [03 04]⟩ ⟨119886 [03 04]⟩

⟨119887 [03 04]⟩ ⟨119888 [03 08]⟩ ⟨0 [05 09]⟩

(14)

Theorem 27 Let 119860 be a vague set of a residuated lattice 119871Then the vague value of 119909 in ⟨119860⟩ is

119881⟨119860⟩ (119909) = isup imin VA (a1) VA (a2) VA (an)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

(15)

for all 119909 isin 119871

Proof First we will prove that ⟨119860⟩ is a vague set of 119871

119905⟨119860⟩ (119909)

= sup min 119905119860(1198861) 119905119860(1198862) 119905

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

le sup min 1 minus 119891119860(1198861) 1 minus 119891

119860(1198862) 1 minus 119891

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

= sup 1 minus max 119891119860(1198861) 119891119860(1198862) 119891

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

= 1 minus inf max 119891119860(1198861) 119891119860(1198862) 119891

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

= 1 minus 119891⟨119860⟩ (119909)

(16)

Now we will show that ⟨119860⟩ is a vague filter of 119871(VF1) Suppose that 119909 119910 isin 119871 is arbitrary For every 120576 gt 0

we can take 1198861 119886

119899 1198871 119887

119898isin 119871 such that119909 ge 119886

1lowastsdot sdot sdotlowast119886

119899

119910 ge 1198871lowast sdot sdot sdot lowast 119887

119898 and

119905⟨119860⟩ (119909) minus 120576 lt min 119905

119860(1198861) 119905119860(1198862) 119905

119860(119886119899)

119905⟨119860⟩

(119910) minus 120576 lt min 119905119860(1198871) 119905119860(1198872) 119905

119860(119887119898)

(17)

By Proposition 2 part (4) we have 119909 lowast 119910 ge 1198861lowast sdot sdot sdot lowast 119886

119899lowast 1198871lowast

sdot sdot sdot lowast 119887119898and then

119905⟨119860⟩

(119909 lowast 119910)

ge min 119905119860(1198861) 119905

119860(119886119899) 119905119860(1198871) 119905

119860(119887119898)

= min min 119905119860(1198861) 119905

119860(119886119899)

min 119905119860(1198871) 119905

119860(119887119898)

gt min (119905⟨119860⟩ (119909) minus 120576) (119905

⟨119860⟩(119910) minus 120576)

= min 119905⟨119860⟩ (119909) 119905⟨119860⟩ (119910) minus 120576

(18)

Since 120576 gt 0 is arbitrary we obtain that 119905⟨119860⟩

(119909 lowast 119910) ge

min119905⟨119860⟩

(119909) 119905⟨119860⟩

(119910)Similarly we can show that 1 minus 119891

⟨119860⟩(119909 lowast 119910) ge min1 minus

119891⟨119860⟩

(119909) 1 minus119891⟨119860⟩

(119910) Hence119881119860(119909lowast119910) ge imin119881

119860(119909) 119881119860(119910)

for all 119909 119910 isin 119871(VF2) Suppose that 119909 le 119910where 119909 119910 isin 119871 Let 119886

1 119886

119899isin

119871 such that 1198861lowast sdot sdot sdot lowast 119886

119899le 119909 Then 119886

1lowast sdot sdot sdot lowast 119886

119899le 119910 And we

get that

119881119860(119910) ge imin 119881

119860(1198861) 119881119860(1198862) 119881

119860(119886119899) (19)

Since 1198861 119886

119899isin 119871 where 119909 ge 119886

1lowast sdot sdot sdot lowast 119886

119899is arbitrary then

119881⟨119860⟩

(119910) ge isup imin 119881119860(1198861) 119881119860(1198862) 119881

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

= 119881⟨119860⟩ (119909)

(20)

Therefore ⟨119860⟩ is a vague filter of 119871 by Theorem 16 It is clearthat119860 ⪯ ⟨119860⟩ Now let119862 be a vague filter of119871 such that119860 ⪯ 119862Then we have

119881⟨119860⟩ (119909)

= isup imin 119881119860(1198861) 119881

119860(119886119899) 119909 ge 119886

1lowast sdot sdot sdot lowast 119886

119899

le isup imin 119881119862(1198861) 119881

119862(119886119899) 119909 ge 119886

1lowast sdot sdot sdot lowast 119886

119899

= 119881119862 (119909)

(21)

for all 119909 isin 119871 Hence ⟨119860⟩ ⪯ 119862 and then ⟨119860⟩ is a vague filtergenerated by 119860

We denote the set of all vague filters of a residuated lattice119871 by VF(119871)

6 Journal of Discrete Mathematics

Corollary 28 (VF(119871) cap or) is a complete lattice

Proof Clearly if 119860119894119894isinΓ

is a family of vague filters of aresiduated lattice 119871 then the infimumof this family is⋂

119894isinΓ119860119894

and the supremum is ⋁119894isinΓ

119860119894= ⟨⋃119894isinΓ

119860119894⟩ It is easy to prove

that (IFF(119871) cap or) is a complete lattice

Let 119860 and 119861 be vague sets of a residuated lattice 119871 Then119860 lowast 119861 is defined as follows

119881(119860lowast119861) (119911) = isup imin 119881

119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910 (22)

for all 119911 isin 119871

Theorem 29 Let 119860 and 119861 be vague sets of a residuated lattice119871 Then 119860 lowast 119861 is a vague set of 119871

Proof Since 119860 and 119861 are vague sets of 119871 we have 119905119860(119909) le

1 minus 119891119860(119909) and 119905

119861(119910) le 1 minus 119891

119861(119910) for all 119909 119910 isin 119871 Thus

119905119860lowast119861 (119911) = sup min 119905

119860 (119909) 119905119860 (119910) 119911 ge 119909 lowast 119910

le sup min 1 minus 119891119860 (119909) 1 minus 119891

119860(119910) 119911 ge 119909 lowast 119910

le 1 minus 119891119860lowast119861 (119911)

(23)

Hence 119905119860lowast119861

(119911) + 119891119860lowast119861

(119911) le 1 and then 119860 lowast 119861 is a vague set of119871

Theorem30 Let119860 and119861 be vague filters of a residuated lattice119871 Then 119860 lowast 119861 is a vague filter of 119871

Proof ByTheorem 29119860lowast119861 is a vague set of 119871 We will provethat 119860 lowast 119861 is a vague filter of 119871

(VF2) Let 1199111

le 1199112and 119909 lowast 119910 le 119911

1 We get that

imin119881119860(119909) 119881119861(119910) le 119881

119860lowast119861(1199112) Since 119909 lowast 119910 le 119911

1is arbitrary

we have

119881119860lowast119861

(1199111) = isup imin 119881

119860 (119909) 119881119861 (119910) 1199111ge 119909 lowast 119910

le 119881119860lowast119861

(1199112)

(24)

(VF1) Let 119909 119910 isin 119871 Then

imin 119881119860lowast119861 (119909) 119881119860lowast119861 (119910)

= imin isup imin 119881119860 (119886) 119881119861 (119887) 119909 ge 119886 lowast 119887

isup imin 119881119860 (119888) 119881119861 (119889) 119910 ge 119888 lowast 119889

= isup isup imin 119881119860 (119886 lowast 119888) 119881119861 (119888 lowast 119889)

119909 ge 119886 lowast 119887 119910 ge 119888 lowast 119889

le isup imin 119881119860 (119886 lowast 119888) 119881119861 (119888 lowast 119889)

119909 lowast 119910 ge (119886 lowast 119887) lowast (119888 lowast 119889)

le isup imin 119881119860 (119890) 119881119861 (119891) 119909 lowast 119910 ge 119890 lowast 119891

= 119881119860lowast119861

(119909 lowast 119910)

(25)

By Definition 13 119860 lowast 119861 is a vague filter of 119871

Theorem31 Let119860 and119861 be vague filters of a residuated lattice119871 such that119881

119860(1) = 119881

119861(1) Then119860lowast119861 is the least upper bound

of 119860 and 119861

Proof By Definition 13 we have 119881119860(1) = 119881

119861(1) ge 119881

119860(119910) for

all 119910 isin 119871 Let 119911 isin 119871 Since 119911 = 1 lowast 119911 we get that

119881119860lowast119861 (119911) = isup imin 119881

119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910

ge imin 119881119860 (119911) 119881119861 (1) ge 119881

119860 (119911)

(26)

Hence 119860 ⪯ 119860 lowast 119861 Similarly we can show that 119861 ⪯ 119860 lowast 119861Suppose that 119862 is the vague filter of 119871 containing 119860 and 119861Then

119881119860lowast119861 (119911)

= isup imin 119881119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910

le isup imin 119881119862 (119909) 119881119862 (119910) 119911 ge 119909 lowast 119910

le isup imin 119881119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 = 120583

119862 (119911)

(27)

Hence 119860lowast 119861 ⪯ 119862 that is 119860lowast 119861 is the least upper bound of 119860and 119861

Lemma 32 Let119862 be a vague filter of a residuated lattice 119871 and119911 isin 119871 Then

(1) 119881119862(119911) = isupVC(x lowast y) z ge x lowast y for all 119909 119910 119911 isin 119871

(2) 119881119862(119909 lowast 119910) = iminVC(x)VC(y) for 119909 119910 isin 119871

Proof (1) Since 119909 lowast 119910 le 119911 then 119881119862(119909 lowast 119910) le 119881

119862(119911) by (VF2)

Hence 119881119862(119911) ge isup119881

119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 Also we have

1 lowast 119911 = 119911 We obtain that

119881119862 (119911) = 119881

119862 (1 lowast 119911) le isup 119881119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 (28)

(2) It follows from Proposition 2 part (2) andDefinition 13

Let VFD(119871) be the set of all vague filters119860 and119861 of 119871 suchthat 119881

119860(1) = 119881

119861(1)

Theorem 33 (119881119865119863(119871) cap lowast) is a distributive lattice

Journal of Discrete Mathematics 7

Proof Let 119860 119861 119862 isin VFD(119871) be such that 119860 ⪯ 119862 We willshow that (119860 lowast 119861) cap 119862 = (119860 lowast 119862) cap (119861 lowast 119862) By Lemma 32

119881(119860lowast119861)cap119862 (119911)

= imin (119881119860lowast119861

) (119911) 119881119862 (119911)

= imin isup imin 119881119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910 119881

119862 (119911)

= isup imin 119881119860 (119909) 119881119861 (119910) 119881

119862 (119911) 119911 ge 119909 lowast 119910

= isup imin 119881119860 (119909) 119881119861 (119910) 119881

119862(119909 lowast 119910) 119911 ge 119909 lowast 119910

= isup imin 119881119860 (119909) 119881119861 (119910) 119881

119862 (119909) 119881119862 (119910) 119911 ge 119909 lowast 119910

= isup imin 119881119860 (119909) 119881119862 (119909)

imin 119881119861(119910) 119881

119862(119910) 119911 ge 119909 lowast 119910

= isup imin 119881119860cap119862 (119909) 119881119861cap119862 (119910) 119911 ge 119909 lowast 119910

= 119881(119860cap119862)lowast(119861cap119862) (119911)

(29)

5 Subpositive Implicative Vague Filters

Definition 34 Let 119860 be a vague subset of a residuated lattice119871 119860 is called a subpositive implicative vague filter of 119871 if

(VF6) 119881119860(119909) le 119881

119860(1) for all 119909 isin 119871

(SVF1) imin119881119860(((119909 rarr 119910) lowast 119911) 999492999484 ((119910 999492999484 119909) rarr

119909)) 119881119860(119911) le 119881

119860((119909 rarr 119910) 999492999484 119910) for any

119909 119910 119911 isin 119871(SVF2) imin119881

119860((119911 lowast (119909 999492999484 119910)) rarr ((119910 rarr 119909) 999492999484

119909)) 119881119860(119911) le 119881

119860((119909 999492999484 119910) rarr 119910) for any

119909 119910 119911 isin 119871

Proposition 35 Let 119860 be a vague filter of a residuated lattice119871 Then 119860 is a subpositive implicative vague filter of L if andonly if it satisfies

(SVF3) 119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) le

119881119860((119909 rarr 119910) 999492999484 119910) for any 119909 119910 isin 119871

(SVF4) 119881119860((119909 999492999484 119910) rarr ((119910 rarr 119909) 999492999484 119909)) le

119881119860((119909 999492999484 119910) rarr 119910) for any 119909 119910 isin 119871

Proposition 36 Let119860 be a subpositive implicative vague filterof a residuated lattice 119871 Then 119860 is a vague filter of 119871

Proof We have imin119881119860(119909 999492999484 119910) 119881

119860(119909) = imin119881

119860((119910 rarr

119910) lowast 119909) 999492999484 ((119910 999492999484 119910) rarr 119910) 119881119860(119909) le 119881

119860((119910 rarr 119910) 999492999484

119910) = 119881119860(119910) By (VS1) andTheorem 17 119860 is a vague filter of a

residuated lattice 119871

The relation between subpositive implicative vague filterand its level subset is as follows

Theorem 37 Let 119860 be a vague set of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and only if

for all120572 120573 isin [0 1] the set119860(120572120573)

is either empty or a subpositiveimplicative filter of 119871 where 120572 le 120573

Proof Suppose that119860 is a subpositive implicative filter vagueset of 119871 and 120572 120573 isin [0 1] Let 119860

(120572120573)= 0 Since 119860 is a vague

filter then119860(120572120573)

is a filter of 119871 byTheorem 21 Now let (119909 rarr

119910) 999492999484 ((119910 999492999484 119909) rarr 119909) isin 119860(120572120573)

Then 119881119860((119909 rarr 119910) 999492999484

((119910 999492999484 119909) rarr 119909)) ge [120572 120573] By (SVF3) 119881119860((119909 rarr 119910) 999492999484

119910) ge [120572 120573] that is (119909 rarr 119910) 999492999484 119910 isin 119860(120572120573)

Similarly we canshow that119860

(120572120573)satisfies (SVF4) Hence119860

(120572120573)is a subpositive

implicative filter of 119871Conversely let 119905

119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) =

120572 and 1 minus 119891119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) = 120573 Thus

119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) ge [120572 120573] We get that

(119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909) isin 119860(120572120573)

So 119860(120572120573)

= 0Thus (119909 rarr 119910) 999492999484 119910 isin 119860

(120572120573) that is 119881

119860((119909 rarr 119910) 999492999484 119910) ge

[120572 120573] = 119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) Similarly we can

prove that 119860 satisfies (SVF4)

In the following theorems we obtain some characteriza-tions of subpositive implicative vague filter

Theorem 38 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF5) iminVA((y rarr z) 999492999484 (x rarr y))VA(x) le

VA(y) for any 119909 119910 119911 isin 119871(SVF6) iminVA((y 999492999484 z) rarr (x 999492999484 y))VA(x) le

VA(y) for any 119909 119910 119911 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871 ByProposition 2 part (3) we have imin119881

119860((119910 rarr 119911) 999492999484 (119909 rarr

119910)) 119881119860(119909) = imin119881

119860(119909 rarr ((119910 rarr 119911) 999492999484 119910)) 119881

119860(119909) le

119881119860((119910 rarr 119911) 999492999484 119910) By Proposition 2 part (7) (119910 rarr 119911) 999492999484

119910 le 119911 999492999484 119910 le (119910 rarr 119911) 999492999484 ((119911 999492999484 119910) rarr 119910) We get that119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119911 999492999484 119910) le 119881

119860((119910 rarr 119911) 999492999484 ((119911 999492999484

119910) rarr 119910)) le 119881119860((119911 999492999484 119910) rarr 119910) Hence imin119881

119860((119910 rarr

119911) 999492999484 119910) 119881119860(119911 999492999484 119910) le imin119881

119860((119911 999492999484 119910) rarr 119910) 119881

119860(119911 999492999484

119910) le 119881119860(119910) Thus it satisfies (SVF5) Similarly we can

prove (SVF6) Conversely let 119860 satisfy (SVF5) and (SVF6)119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) = imin119881

119860([((119909 rarr 119910) 999492999484

119910) rarr 1] 999492999484 [(119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909) rarr ((119909 rarr

119910) 999492999484 119910)]) 119881119860(1) le 119881

119860((119909 rarr 119910) 999492999484 119910) Similarly we

can prove that (SVF5) holds Hence 119860 is a vague subpositiveimplicative filter

Theorem 39 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF7) 119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) le

119881119860((119910 999492999484 119909) rarr 119909) for any 119909 119910 isin 119871

(SVF8) 119881119860((119909 999492999484 119910) rarr ((119910 999492999484 119909) rarr 119909)) le

119881119860((119910 rarr 119909) 999492999484 119909) for any 119909 119910 isin 119871

Proof Let (SVF7) and (SVF8) hold We will show that 119860

satisfies (SVF5) and (SVF6) We have imin119881119860((119910 rarr 119911) 999492999484

(119909 rarr 119910)) 119881119860(119909) = imin119881

119860(119909 rarr ((119910 rarr 119911) 999492999484

119910)) 119881119860(119909) le 119881

119860((119910 rarr 119911) 999492999484 119910) le 119881

119860((119910 rarr 119911) 999492999484

8 Journal of Discrete Mathematics

((119911 rarr 119910) 999492999484 119910)) le 119881119860((119911 999492999484 119910) rarr 119910)) Also we have

119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119911 999492999484 119910) Since 119860 is a vague

filter 119881119860((119910 rarr 119911) 999492999484 119910) le imin119881

119860(119911 999492999484 119910) 119881

119860((119911 999492999484

119910) rarr 119910) le 119881119860(119910) Hence 119860 satisfies (SVF5) Analogously

we can prove that119860 satisfies (SVF6) Hence119860 is a subpositiveimplicative vague filter of 119871 byTheorem 38 Conversely let 119860be a subpositive implicative vague filter of 119871 We have

[((119910 999492999484 119909) 997888rarr 119909) 997888rarr 1]

999492999484 [((119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909))

997888rarr ((119910 999492999484 119909) 997888rarr 119909) ]

= [(119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)]

997888rarr [((119910 997888rarr 119909) 999492999484 119909) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)]

= (119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)

(30)

We get that 119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) =

imin119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909))119881

119860([((119910 999492999484 119909) rarr

119909) rarr 1] 999492999484 [((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) rarr ((119910 999492999484

119909) rarr 119909)]) le 119881119860((119910 999492999484 119909) rarr 119909) by (SVF5) Similarly we

can prove that (SVF8) holds

Theorem 40 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF9) 119881119860((119909 rarr 119910) 999492999484 119909) le 119881

119860(119909) for any 119909 119910 isin 119871

(SVF10) 119881119860((119909 999492999484 119910) rarr 119909) le 119881

119860(119909) for any 119909 119910 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871Then 119860 satisfies (SVF5) and (SVF6)

119881119860((119909 997888rarr 119910) 999492999484 119909)

= imin 119881119860(1 997888rarr ((119909 997888rarr 119910) 999492999484 119909)) 119881

119860 (1)

= imin 119881119860((119909 997888rarr 119910) 999492999484 (1 997888rarr 119909)) 119881119860 (1)

le 119881119860 (119909)

(31)

Similarly we can prove that (SVF10) holds Conversely let119860 satisfy (SVF9) and (SVF10) We will show that 119860 satisfies(SVF5) and (SVF6) imin119881

119860((119910 rarr 119911) 999492999484 (119909 rarr

119910)) 119881119860(119909) = imin119881

119860((119909 rarr (119910 rarr 119911) 999492999484 119910)) 119881

119860(119909) le

119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119910) Hence 119860 satisfies (SVF5)

Similarly we can prove (SVF7) holds Therefore 119860 is asubpositive implicative vague filter of L byTheorem 38

Theorem 41 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF11) 119881119860(not119909 999492999484 119909) le 119881

119860(119909) for any 119909 isin 119871

(SVF12) 119881119860(sim 119909 rarr 119909) le 119881

119860(119909) for any 119909 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871Then (SVF11) and (SVF12) follow by (SVF9) and (SVF10)Conversely since 0 le 119910 then 119909 rarr 0 le 119909 rarr 119910 Hence

(119909 rarr 119910) 999492999484 119909 le not119909 999492999484 119909 by Proposition 2 part (6)Therefore 119881

119860((119909 rarr 119910) 999492999484 119909) le 119881

119860(not119909 999492999484 119909) le 119881

119860(119909)

Similarly we can prove that (SVF10) holds Hence 119860 is asubpositive implicative vague filter of 119871 byTheorem 40

The extension theorem of subpositive implicative vaguefilters is obtained from the following theorem

Theorem 42 Let 119860 and 119861 be two vague filters of a residuatedlattice 119871 which satisfies 119860 le 119861 and 119881

119860(1) = 119881

119861(1) If 119860 is a

subpositive implicative vague filter of L so 119861 is

Proof Suppose that 119906 = (119909 rarr 119910) 999492999484 119909 Then

1 = 119906 997888rarr 119906 = 119906 997888rarr ((119909 997888rarr 119910) 999492999484 119909)

= (119909 997888rarr 119910) 999492999484 (119906 997888rarr 119909) = 1

(32)

We have (119909 rarr 119910) 999492999484 (119906 rarr 119909) le ((119906 rarr 119909) rarr 119910) 999492999484

(119906 rarr 119909) Therefore 119881119860(((119906 rarr 119909) rarr 119910) 999492999484 (119906 rarr 119909)) =

119881119860(1) By (SVF10) 119881

119860(1) = 119881

119860(((119906 rarr 119909) rarr 119910) 999492999484 (119906 rarr

119909)) le 119881119860(119906 rarr 119909) Hence 119881

119860(119906 rarr 119909) = 119881

119860(1) = 119881

119861(1)

Since 119860 le 119861 then 119881119861(119906 rarr 119909) ge 119881

119860(119906 rarr 119909) = 119881

119861(1)

We get that 119881119861((119909 rarr 119910) 999492999484 119909) = 119881

119861(119906) = imin119881

119861(119906 rarr

119909) 119881119861(119906) le 119881

119861(119909) Similarly we can show that 119866 satisfies

(SVF11) Hence 119866 is a vague subpositive implicative filter of119871 by Theorem 40

Definition 43 Let 119860 be a vague filter of a residuated lattice 119871119860 is called a vague Boolean filter of 119871 if

(BVF1) 119881119860(119909 or not119909) = 119881

119860(1) for all 119909 isin 119871

(BVF2) 119881119860(119909or sim 119909) = 119881

119860(1) for all 119909 isin 119871

Similar to Theorem 40 we have the following theoremwith respect to Boolean vague filters

Theorem 44 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a Boolean vague filter of 119871 if and only if for all120572 120573 isin [0 1] the set 119860

(120572120573)is either empty or a Boolean filter of

119871 where 120572 le 120573

Theorem 45 Let 119860 be a vague filter of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and if onlyif 119860 is a Boolean vague filter

Proof Let119860 be a subpositive implicative vague filterWe have

119881119860 (1) = 119881

119860 (not (119909 or not119909) 999492999484 (119909 or not119909))

= 119881119860 ((not119909 and notnot119909) 999492999484 (119909 or not119909)) le 119881

119860 (119909 or not119909)

(33)

Hence 119881119860(119909 or not119909) = 119881

119860(1) Similarly we can prove that 119860

satisfies (BVF2) Hence 119860 is a Boolean vague filterConversely let119881

119860(119909ornot119909) = 119881

119860(1)We have119881

119860(119909ornot119909) =

119881119860((119909 999492999484 119909) or (not119909 999492999484 119909))119881

119860((119909 or not119909) 999492999484 119909) = imin119881

119860((119909 or

not119909) 999492999484 119909) 119881119860(1) = imin119881

119860((119909 or not119909) 999492999484 119909) 119881

119860(119909 or not119909) le

119881119860(119909) Similarly we can show that119860 satisfies (SVF12) Hence

119860 is a subpositive implicative vague filter byTheorem 41

Journal of Discrete Mathematics 9

6 Conclusion

In this paper we introduced the notions of vague filterssubpositive implicative vague filters and Boolean vaguefilters of a residuated lattice and investigated some relatedproperties We studied the relationship between filters andvague filters of a residuated lattice and showed that vaguefilters of a residuated lattice is a generalization of fuzzy filtersWe proved that the set of all vague filters of a residuated latticeforms a complete lattice and obtained some characterizationsof subpositive implicative vague filterThe extension theoremof subpositive implicative vague filters was obtained Finallyit was proved that subpositive implicative vague filters areequivalent to Boolean vague filters

Conflict of Interests

There is no conflict of interests regarding the publication ofthis paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993

[3] Y B Jun and C H Park ldquoIdeals of pseudo MV-algebras basedon vague set theoryrdquo Iranian Journal of Fuzzy Systems vol 6 no7 pp 31ndash47 2009

[4] A Borumand Saeid ldquoVague BCKBCI-algebrasrdquo OpusculaMathematica vol 29 no 2 pp 177ndash186 2009

[5] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939

[6] T S Blyth and M F Janowitz Residuation Theory PergamonPress 1972

[7] T Kowalski and H Ono Residuated Lattices An AlgebraicGlimpse at Logic without Contraction Japan Advanced Instituteof Science and Technology March 2001

[8] D PiciuAlgebras of Fuzzy Logic Editura Universitaria CraiovaCraiova Romania 2007

[9] E TurunenMathematics Behind Fuzzy Logic Physica 1999[10] P Hajek Metamathematics of Fuzzy Logic vol 4 of Trends

in LogicmdashStudia Logica Library Kluwer Academic DordrechtThe Netherlands 1998

[11] Sh Ghorbani ldquoSub positive implicative filters of residuatedlatticesrdquo World Applied Science Journal vol 12 no 5 pp 586ndash590 2011

[12] S Ghorbani and A Hasankhani ldquoFuzzy convex subalgebrasof commutative residuated latticesrdquo Iranian Journal of FuzzySystems vol 7 no 2 pp 41ndash54 159 2010

[13] R Biswas ldquoVague groupsrdquo International Journal of Computa-tional Cognition vol 4 pp 20ndash23 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Discrete Mathematics 3

an order le on elements 1198681= [1198861 1198871] and 119868

2= [1198862 1198872] of 119868[0 1]

asimin (119868

1 1198682) = [min 119886

1 1198862 min 119887

1 1198872]

1198681le 1198682

iff 1198861le 1198862 1198871le 1198872

(1)

Similarly we can define ge = and imax Then the concept ofimin and imax could be extended to define iinf and isup ofinfinite number of elements of 119868[0 1] It is a known fact that119871 = 119868[0 1] iinf isup le is a lattice with universal boundszero vague set and unit vague set For 120572 120573 isin [0 1] we nowdefine (120572 120573)-cut and 120572-cut of a vague set

Definition 11 (see [13]) Let 119860 be a vague set of a universe 119883

with the true-membership function 119905119860and false-membership

function119891119860The (120572 120573)-cut of the vague set119860 is a crisp subset

119860(120572120573)

of the set 119883 given by

119860(120572120573)

= 119909 isin 119883 119881119860 (119909) ge [120572 120573]

where 120572 le 120573

(2)

Clearly 119860(00)

= 119883 The (120572 120573)-cuts are also called vague-cutsof the vague set 119860

Definition 12 (see [13]) The 120572-cut of the vague set119860 is a crispsubset 119860

120572of the set119883 given by 119860

120572= 119860(120572120572)

Equivalently wecan define the 120572-cut as 119860

120572= 119909 isin 119883 119905

119860(119909) ge 120572

3 Vague Filters of Residuated Lattices

In this section we define the notion of vague filters ofresiduated lattices and obtain some related results

Definition 13 A vague set 119860 of a residuated lattice 119871 is calleda vague filter of 119871 if the following conditions hold

(VF1) if 119909 le 119910 then 119881119860(119909) le 119881

119860(119910) for all 119909 119910 isin 119871

(VF2) imin119881119860(119909) 119881119860(119910) le 119881

119860(119909lowast119910) for all 119909 119910 isin

119871

That is(1) 119909 le 119910 implies 119905

119860(119909) le 119905

119860(119910) and 1minus119891

119860(119909) le 1minus119891

119860(119910)

for all 119909 119910 isin 119871(2) min119905

119860(119909) 119905119860(119910) le 119905

119860(119909 lowast 119910) and min1 minus 119891

119860(119909) 1 minus

119891119860(119910) le 1 minus 119891

119860(119909 lowast 119910) for all 119909 119910 isin 119871

In the following proposition we will show that the vaguefilters of a residuated lattice exist

Proposition 14 Unit vague set and120572-vague set of a residuatedlattice 119871 are vague filters of 119871

Proof Let 119860 be a 120572-vague set of 119871 It is clear that 119860 satisfies(VF1) For 119909 119910 isin 119871 we have

119905119860(119909 lowast 119910) = 120572 = min 120572 120572 = min 119905

119860 (119909) 119905119860 (119910)

1 minus 119891119860(119909 lowast 119910)

= 120572 = min 120572 120572 = min 1 minus 119891119860 (119909) 1 minus 119891

119860(119910)

(3)

Thus imin119881119860(119909) 119881119860(119910) le 119881

119860(119909 lowast 119910) for all 119909 119910 isin 119871 Hence

it is a vague filter of 119871 The proof of the other cases is similar

Lemma 15 Let 119860 be a vague set of a residuated lattice 119871 suchthat it satisfies (VF1) Then the following are equivalent

(VF4) iminVA(x)VA(x rarr y) le VA(y)(VF5) iminVA(x)VA(x 999492999484 y) le VA(y)

for all 119909 119910 isin 119871

Proof Suppose that 119860 satisfies (VF4) We have 119909 le (119909 999492999484

119910) rarr 119910 By (VF1) 119881119860(119909) le 119881

119860((119909 999492999484 119910) rarr 119910) We get that

imin 119881119860 (119909) 119881119860 (119909 999492999484 119910)

le imin 119881119860((119909 999492999484 119910) 997888rarr 119910) 119881

119860(119909 999492999484 119910)

le 119881119860(119910)

(4)

Conversely if119860 satisfies (VF5) then the inequality 119909 le (119909 rarr

119910) 999492999484 119910 analogously entails (VF4)

Theorem 16 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter if and only if it satisfies (VF4) and

(VF6) 119881119860(119909) le 119881

119860(1) for all 119909 isin 119871

Proof Let 119860 be a vague filter of 119871 Since 119909 le 1 then 119881119860(119909) le

119881119860(1) for all 119909 isin 119871 by (VF1) We have 119909 lowast (119909 rarr 119910) le 119910 By

(VF1) and (VF2) we get

imin 119881119860 (119909) 119881119860 (119909 997888rarr 119910)

le 119881119860(119909 lowast (119909 997888rarr 119910)) le 119881

119860(119910)

(5)

Hence (VF4) holdConversely let 119860 satisfy (VF4) and (VF6) Since 119909 le 119910

then 119909 rarr 119910 = 1

119881119860(119910) ge imin 119881

119860(119909 997888rarr 119910) 119881

119860 (119909)

= imin 119881119860 (1) 119881119860 (119909) = 119881

119860 (119909)

(6)

Hence 119881119860

is order-preserving and (VF1) holds ByProposition 2 part (10) 119909 rarr (119910 999492999484 (119909 lowast 119910)) = 1 By(VF4) we have

119881119860(119910 999492999484 (119909 lowast 119910))

ge imin 119881119860 (119909) 119881119860 (119909 997888rarr (119910 999492999484 (119909 lowast 119910))) = 119881

119860 (119909)

(7)

By Lemma 15

119881119860(119909 lowast 119910) ge imin 119881

119860(119910 999492999484 (119909 lowast 119910)) 119881

119860(119910)

ge imin 119881119860 (119909) 119881119860 (119910)

(8)

Hence (VF2) holds

Theorem 17 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter if and only if it satisfies (VF5) and(VF6)

4 Journal of Discrete Mathematics

Definition 18 A vague set 119860 of a residuated lattice 119871 is calleda vague lattice filter of 119871 if it satisfies (VF1) and

(VLF2) imin119881119860(119909) 119881119860(119910) le 119881

119860(119909 and 119910) for all

119909 119910 isin 119871

Proposition 19 Let 119860 be a vague filter of a residuated lattice119871 then 119860 is a vague lattice filter

In the following example we will show that the converseof Proposition 19 may not be true in general

Example 20 Let 0 119886 119887 119888 1 such that 0 lt 119886 lt 119887 119888 lt 1where 119887 and 119888 are incomparable Define the operations lowast rarr and 999492999484 as follows

lowast 0 119886 119887 119888 1

0 0 0 0 0 0

119886 0 0 0 119886 119886

119887 0 119886 119887 119886 119887

119888 0 0 0 119888 119888

1 0 119886 119887 119888 1

997888rarr 0 119886 119887 119888 1

0 1 1 1 1 1

119886 119888 1 1 1 1

119887 119888 119888 1 119888 1

119888 0 119887 119887 1 1

1 0 119886 119887 119888 1

999492999484 0 119886 119887 119888 1

0 1 1 1 1 1

119886 119887 1 1 1 1

119887 0 119888 1 119888 1

119888 119887 119887 119887 1 1

1 0 119886 119887 119888 1

(9)

Then 119871 is a residuated lattice Let 119860 be the vague set definedas follows

119860 = ⟨0 [01 02]⟩ ⟨119886 [03 04]⟩ ⟨119887 [03 04]⟩

⟨119888 [03 08]⟩ ⟨0 [05 09]⟩

(10)

It is routine to verify that119860 is a vague lattice filter but it is nota vague filter of 119871

Let 119861 be the vague set defined as follows

119861 = ⟨0 [02 05]⟩ ⟨119886 [02 05]⟩ ⟨119887 [02 05]⟩

⟨119888 [04 07]⟩ ⟨0 [07 07]⟩

(11)

Then 119861 is a vague filter of 119871

In the following theorems we will study the relationshipbetween filters and vague filters of a residuated lattice

Theorem 21 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter of 119871 if and only if for all 120572 120573 isin [0 1]the set 119860

(120572120573)is either empty or a filter of 119871 where 120572 le 120573

Proof Suppose that 119860 is a vague filter of 119871 and 120572 120573 isin [0 1]Let 119860

(120572120573)= 0 We will show that 119860

(120572120573)is a filter of 119871

(1) Suppose that 119909 le 119910 and 119909 isin 119860(120572120573)

By (VF1)119881119860(119910) ge

119881119860(119909) ge [120572 120573] Therefor 119910 isin 119860

(120572120573)

(2) Suppose that 119909 119910 isin 119860(120572120573)

Then 119881119860(119909) 119881119860(119910) ge

[120572 120573] By (VF2) we have 119881119860(119909 lowast 119910) ge

imin119881119860(119909) 119881119860(119910) ge [120572 120573] Hence 119909 lowast 119910 isin 119860

(120572120573)

and then 119860(120572120573)

is a filter of 119871

Conversely suppose that for all 120572 120573 isin [0 1] the sets 119860(120572120573)

is either empty or a filter of 119871 Let 119905119860(119909) = 120572

1 119905119860(119910) = 120572

2

1 minus 119891119860(119909) = 120573

1 and 1 minus 119891

119860(119910) = 120573

2 Put 120572 = min120572

1 1205722

and 120573 = min1 minus 1205731 1 minus 120573

2 Then 119905

119860(119909) 119905119860(119910) ge 120572 and 1 minus

119891119860(119909) 1 minus 119891

119860(119910) ge 120573 Hence 119881

119860(119910) 119881119860(119909) ge [120572 120573] that is

119909 119910 isin 119860(120572120573)

So 119860(120572120573)

= 0 and by assumption 119860(120572120573)

is afilter of 119871 We obtain that 119909 lowast 119910 isin 119860

(120572120573) that is

119881119860(119909 lowast 119910) ge [120572 120573]

= [min 1205721 1205722 min 1 minus 120573

1 1 minus 120573

2]

= imin [119905119860 (119909) 1 minus 119891

119860 (119909)] [119905119860 (119910) 1 minus 119891119860(119910)]

= imin 119881119860 (119909) 119881119860 (119910)

(12)

Hence (VF2) holdsSuppose that 119909 le 119910 119905

119860(119909) = 120572 and 1 minus 119891

119860(119909) = 120573 Hence

119909 isin 119860(120572120573)

By assumption 119860(120572120573)

is a filter We get that 119910 isin

119860(120572120573)

that is 119881119860(119910) ge [120572 120573] = [119905

119860(119909) 1 minus 119891

119860(119909)] = 119881

119860(119909)

Hence 119860 is a vague filter of 119871

The filters like119860(120572120573)

are also called vague-cuts filters of 119871

Corollary 22 Let 119860 be a vague filter of a residuated lattice 119871Then for all 120572 isin [0 1] the set 119860

120572is either empty or a filter of

119871

Corollary 23 Any filter 119865 of a residuated lattice 119871 is a vague-cut filter of some vague filter of 119871

Proof For all 119909 isin 119871 define

1 minus 119891119860 (119909) = 119905

119860 (119909) = 1 if 119909 isin 119865

120572 otherwise(13)

Hence we have 119881119860(119909) = [1 1] if 119909 isin 119865 and 119881

119860(119909) = [120572 120572]

otherwise where 120572 isin (0 1) It is clear that 119865 = 119860(11)

Let119909 119910 isin 119871 If 119909 119910 isin 119865 then 119909 lowast 119910 isin 119865 We have 119881

119860(119909 lowast 119910) =

[1 1] = imin119881119860(119909) 119881119860(119910) If one of 119909 or 119910 does not belong

to 119865 then one of119881119860(119909) and119881

119860(119910) is equal to [120572 120572]Therefore

119881119860(119909lowast119910) ge [120572 120572] = imin119881

119860(119909) 119881119860(119910) Suppose that 119909 le 119910

If 119909 isin 119865 then 119910 isin 119865 Hence 119881119860(119909) = 119881

119860(119910) If 119909 notin 119865 then

119881119860(119909) = [120572 120572] We have 119881

119860(119909) le 119881

119860(119910) Hence 119860 is a vague

filter of 119871

In the following proposition we will show that vaguefilters of a residuated lattice are a generalization of fuzzyfilters

Proposition 24 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter of 119871 if and only if the fuzzy sets 119905

119860and

1 minus 119891119860are fuzzy filters in 119871

Journal of Discrete Mathematics 5

4 Lattice of Vague Filters

Let 119860 and 119861 be vague sets of a residuated lattice 119871 If 119881119860(119909) le

119881119861(119909) for all 119909 isin 119871 then we say 119861 containing119860 and denote it

by 119860 ⪯ 119861The unite vague set of a residuated lattice 119871 is a vague

filter of 119871 containing any vague filter of 119871 For a vague setwe can define the intersection of two vague sets 119860 and 119861 of aresiduated lattice 119871 by 119881

119860cap119861(119909) = imin119881

119860(119909) 119881119861(119909) for all

119909 isin 119871 It is easy to prove that the intersection of any family ofvague filters of 119871 is a vague filter of 119871 Hence we can define avague filter generated by a vague set as follows

Definition 25 Let 119860 be a vague set of a residuated lattice 119871A vague filter 119861 is called a vague filter generated by 119860 if itsatisfies

(i) 119860 ⪯ 119861(ii) 119860 ⪯ 119862 that implies 119861 ⪯ 119862 for all vague filters 119862 of 119871

The vague filter generated by 119860 is denoted by ⟨119860⟩

Example 26 Consider the vague lattice filter119860 in Example 20which is not a vague filter of 119871Then the vague filter generatedby 119860 is

⟨119860⟩ = ⟨0 [03 04]⟩ ⟨119886 [03 04]⟩

⟨119887 [03 04]⟩ ⟨119888 [03 08]⟩ ⟨0 [05 09]⟩

(14)

Theorem 27 Let 119860 be a vague set of a residuated lattice 119871Then the vague value of 119909 in ⟨119860⟩ is

119881⟨119860⟩ (119909) = isup imin VA (a1) VA (a2) VA (an)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

(15)

for all 119909 isin 119871

Proof First we will prove that ⟨119860⟩ is a vague set of 119871

119905⟨119860⟩ (119909)

= sup min 119905119860(1198861) 119905119860(1198862) 119905

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

le sup min 1 minus 119891119860(1198861) 1 minus 119891

119860(1198862) 1 minus 119891

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

= sup 1 minus max 119891119860(1198861) 119891119860(1198862) 119891

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

= 1 minus inf max 119891119860(1198861) 119891119860(1198862) 119891

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

= 1 minus 119891⟨119860⟩ (119909)

(16)

Now we will show that ⟨119860⟩ is a vague filter of 119871(VF1) Suppose that 119909 119910 isin 119871 is arbitrary For every 120576 gt 0

we can take 1198861 119886

119899 1198871 119887

119898isin 119871 such that119909 ge 119886

1lowastsdot sdot sdotlowast119886

119899

119910 ge 1198871lowast sdot sdot sdot lowast 119887

119898 and

119905⟨119860⟩ (119909) minus 120576 lt min 119905

119860(1198861) 119905119860(1198862) 119905

119860(119886119899)

119905⟨119860⟩

(119910) minus 120576 lt min 119905119860(1198871) 119905119860(1198872) 119905

119860(119887119898)

(17)

By Proposition 2 part (4) we have 119909 lowast 119910 ge 1198861lowast sdot sdot sdot lowast 119886

119899lowast 1198871lowast

sdot sdot sdot lowast 119887119898and then

119905⟨119860⟩

(119909 lowast 119910)

ge min 119905119860(1198861) 119905

119860(119886119899) 119905119860(1198871) 119905

119860(119887119898)

= min min 119905119860(1198861) 119905

119860(119886119899)

min 119905119860(1198871) 119905

119860(119887119898)

gt min (119905⟨119860⟩ (119909) minus 120576) (119905

⟨119860⟩(119910) minus 120576)

= min 119905⟨119860⟩ (119909) 119905⟨119860⟩ (119910) minus 120576

(18)

Since 120576 gt 0 is arbitrary we obtain that 119905⟨119860⟩

(119909 lowast 119910) ge

min119905⟨119860⟩

(119909) 119905⟨119860⟩

(119910)Similarly we can show that 1 minus 119891

⟨119860⟩(119909 lowast 119910) ge min1 minus

119891⟨119860⟩

(119909) 1 minus119891⟨119860⟩

(119910) Hence119881119860(119909lowast119910) ge imin119881

119860(119909) 119881119860(119910)

for all 119909 119910 isin 119871(VF2) Suppose that 119909 le 119910where 119909 119910 isin 119871 Let 119886

1 119886

119899isin

119871 such that 1198861lowast sdot sdot sdot lowast 119886

119899le 119909 Then 119886

1lowast sdot sdot sdot lowast 119886

119899le 119910 And we

get that

119881119860(119910) ge imin 119881

119860(1198861) 119881119860(1198862) 119881

119860(119886119899) (19)

Since 1198861 119886

119899isin 119871 where 119909 ge 119886

1lowast sdot sdot sdot lowast 119886

119899is arbitrary then

119881⟨119860⟩

(119910) ge isup imin 119881119860(1198861) 119881119860(1198862) 119881

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

= 119881⟨119860⟩ (119909)

(20)

Therefore ⟨119860⟩ is a vague filter of 119871 by Theorem 16 It is clearthat119860 ⪯ ⟨119860⟩ Now let119862 be a vague filter of119871 such that119860 ⪯ 119862Then we have

119881⟨119860⟩ (119909)

= isup imin 119881119860(1198861) 119881

119860(119886119899) 119909 ge 119886

1lowast sdot sdot sdot lowast 119886

119899

le isup imin 119881119862(1198861) 119881

119862(119886119899) 119909 ge 119886

1lowast sdot sdot sdot lowast 119886

119899

= 119881119862 (119909)

(21)

for all 119909 isin 119871 Hence ⟨119860⟩ ⪯ 119862 and then ⟨119860⟩ is a vague filtergenerated by 119860

We denote the set of all vague filters of a residuated lattice119871 by VF(119871)

6 Journal of Discrete Mathematics

Corollary 28 (VF(119871) cap or) is a complete lattice

Proof Clearly if 119860119894119894isinΓ

is a family of vague filters of aresiduated lattice 119871 then the infimumof this family is⋂

119894isinΓ119860119894

and the supremum is ⋁119894isinΓ

119860119894= ⟨⋃119894isinΓ

119860119894⟩ It is easy to prove

that (IFF(119871) cap or) is a complete lattice

Let 119860 and 119861 be vague sets of a residuated lattice 119871 Then119860 lowast 119861 is defined as follows

119881(119860lowast119861) (119911) = isup imin 119881

119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910 (22)

for all 119911 isin 119871

Theorem 29 Let 119860 and 119861 be vague sets of a residuated lattice119871 Then 119860 lowast 119861 is a vague set of 119871

Proof Since 119860 and 119861 are vague sets of 119871 we have 119905119860(119909) le

1 minus 119891119860(119909) and 119905

119861(119910) le 1 minus 119891

119861(119910) for all 119909 119910 isin 119871 Thus

119905119860lowast119861 (119911) = sup min 119905

119860 (119909) 119905119860 (119910) 119911 ge 119909 lowast 119910

le sup min 1 minus 119891119860 (119909) 1 minus 119891

119860(119910) 119911 ge 119909 lowast 119910

le 1 minus 119891119860lowast119861 (119911)

(23)

Hence 119905119860lowast119861

(119911) + 119891119860lowast119861

(119911) le 1 and then 119860 lowast 119861 is a vague set of119871

Theorem30 Let119860 and119861 be vague filters of a residuated lattice119871 Then 119860 lowast 119861 is a vague filter of 119871

Proof ByTheorem 29119860lowast119861 is a vague set of 119871 We will provethat 119860 lowast 119861 is a vague filter of 119871

(VF2) Let 1199111

le 1199112and 119909 lowast 119910 le 119911

1 We get that

imin119881119860(119909) 119881119861(119910) le 119881

119860lowast119861(1199112) Since 119909 lowast 119910 le 119911

1is arbitrary

we have

119881119860lowast119861

(1199111) = isup imin 119881

119860 (119909) 119881119861 (119910) 1199111ge 119909 lowast 119910

le 119881119860lowast119861

(1199112)

(24)

(VF1) Let 119909 119910 isin 119871 Then

imin 119881119860lowast119861 (119909) 119881119860lowast119861 (119910)

= imin isup imin 119881119860 (119886) 119881119861 (119887) 119909 ge 119886 lowast 119887

isup imin 119881119860 (119888) 119881119861 (119889) 119910 ge 119888 lowast 119889

= isup isup imin 119881119860 (119886 lowast 119888) 119881119861 (119888 lowast 119889)

119909 ge 119886 lowast 119887 119910 ge 119888 lowast 119889

le isup imin 119881119860 (119886 lowast 119888) 119881119861 (119888 lowast 119889)

119909 lowast 119910 ge (119886 lowast 119887) lowast (119888 lowast 119889)

le isup imin 119881119860 (119890) 119881119861 (119891) 119909 lowast 119910 ge 119890 lowast 119891

= 119881119860lowast119861

(119909 lowast 119910)

(25)

By Definition 13 119860 lowast 119861 is a vague filter of 119871

Theorem31 Let119860 and119861 be vague filters of a residuated lattice119871 such that119881

119860(1) = 119881

119861(1) Then119860lowast119861 is the least upper bound

of 119860 and 119861

Proof By Definition 13 we have 119881119860(1) = 119881

119861(1) ge 119881

119860(119910) for

all 119910 isin 119871 Let 119911 isin 119871 Since 119911 = 1 lowast 119911 we get that

119881119860lowast119861 (119911) = isup imin 119881

119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910

ge imin 119881119860 (119911) 119881119861 (1) ge 119881

119860 (119911)

(26)

Hence 119860 ⪯ 119860 lowast 119861 Similarly we can show that 119861 ⪯ 119860 lowast 119861Suppose that 119862 is the vague filter of 119871 containing 119860 and 119861Then

119881119860lowast119861 (119911)

= isup imin 119881119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910

le isup imin 119881119862 (119909) 119881119862 (119910) 119911 ge 119909 lowast 119910

le isup imin 119881119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 = 120583

119862 (119911)

(27)

Hence 119860lowast 119861 ⪯ 119862 that is 119860lowast 119861 is the least upper bound of 119860and 119861

Lemma 32 Let119862 be a vague filter of a residuated lattice 119871 and119911 isin 119871 Then

(1) 119881119862(119911) = isupVC(x lowast y) z ge x lowast y for all 119909 119910 119911 isin 119871

(2) 119881119862(119909 lowast 119910) = iminVC(x)VC(y) for 119909 119910 isin 119871

Proof (1) Since 119909 lowast 119910 le 119911 then 119881119862(119909 lowast 119910) le 119881

119862(119911) by (VF2)

Hence 119881119862(119911) ge isup119881

119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 Also we have

1 lowast 119911 = 119911 We obtain that

119881119862 (119911) = 119881

119862 (1 lowast 119911) le isup 119881119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 (28)

(2) It follows from Proposition 2 part (2) andDefinition 13

Let VFD(119871) be the set of all vague filters119860 and119861 of 119871 suchthat 119881

119860(1) = 119881

119861(1)

Theorem 33 (119881119865119863(119871) cap lowast) is a distributive lattice

Journal of Discrete Mathematics 7

Proof Let 119860 119861 119862 isin VFD(119871) be such that 119860 ⪯ 119862 We willshow that (119860 lowast 119861) cap 119862 = (119860 lowast 119862) cap (119861 lowast 119862) By Lemma 32

119881(119860lowast119861)cap119862 (119911)

= imin (119881119860lowast119861

) (119911) 119881119862 (119911)

= imin isup imin 119881119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910 119881

119862 (119911)

= isup imin 119881119860 (119909) 119881119861 (119910) 119881

119862 (119911) 119911 ge 119909 lowast 119910

= isup imin 119881119860 (119909) 119881119861 (119910) 119881

119862(119909 lowast 119910) 119911 ge 119909 lowast 119910

= isup imin 119881119860 (119909) 119881119861 (119910) 119881

119862 (119909) 119881119862 (119910) 119911 ge 119909 lowast 119910

= isup imin 119881119860 (119909) 119881119862 (119909)

imin 119881119861(119910) 119881

119862(119910) 119911 ge 119909 lowast 119910

= isup imin 119881119860cap119862 (119909) 119881119861cap119862 (119910) 119911 ge 119909 lowast 119910

= 119881(119860cap119862)lowast(119861cap119862) (119911)

(29)

5 Subpositive Implicative Vague Filters

Definition 34 Let 119860 be a vague subset of a residuated lattice119871 119860 is called a subpositive implicative vague filter of 119871 if

(VF6) 119881119860(119909) le 119881

119860(1) for all 119909 isin 119871

(SVF1) imin119881119860(((119909 rarr 119910) lowast 119911) 999492999484 ((119910 999492999484 119909) rarr

119909)) 119881119860(119911) le 119881

119860((119909 rarr 119910) 999492999484 119910) for any

119909 119910 119911 isin 119871(SVF2) imin119881

119860((119911 lowast (119909 999492999484 119910)) rarr ((119910 rarr 119909) 999492999484

119909)) 119881119860(119911) le 119881

119860((119909 999492999484 119910) rarr 119910) for any

119909 119910 119911 isin 119871

Proposition 35 Let 119860 be a vague filter of a residuated lattice119871 Then 119860 is a subpositive implicative vague filter of L if andonly if it satisfies

(SVF3) 119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) le

119881119860((119909 rarr 119910) 999492999484 119910) for any 119909 119910 isin 119871

(SVF4) 119881119860((119909 999492999484 119910) rarr ((119910 rarr 119909) 999492999484 119909)) le

119881119860((119909 999492999484 119910) rarr 119910) for any 119909 119910 isin 119871

Proposition 36 Let119860 be a subpositive implicative vague filterof a residuated lattice 119871 Then 119860 is a vague filter of 119871

Proof We have imin119881119860(119909 999492999484 119910) 119881

119860(119909) = imin119881

119860((119910 rarr

119910) lowast 119909) 999492999484 ((119910 999492999484 119910) rarr 119910) 119881119860(119909) le 119881

119860((119910 rarr 119910) 999492999484

119910) = 119881119860(119910) By (VS1) andTheorem 17 119860 is a vague filter of a

residuated lattice 119871

The relation between subpositive implicative vague filterand its level subset is as follows

Theorem 37 Let 119860 be a vague set of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and only if

for all120572 120573 isin [0 1] the set119860(120572120573)

is either empty or a subpositiveimplicative filter of 119871 where 120572 le 120573

Proof Suppose that119860 is a subpositive implicative filter vagueset of 119871 and 120572 120573 isin [0 1] Let 119860

(120572120573)= 0 Since 119860 is a vague

filter then119860(120572120573)

is a filter of 119871 byTheorem 21 Now let (119909 rarr

119910) 999492999484 ((119910 999492999484 119909) rarr 119909) isin 119860(120572120573)

Then 119881119860((119909 rarr 119910) 999492999484

((119910 999492999484 119909) rarr 119909)) ge [120572 120573] By (SVF3) 119881119860((119909 rarr 119910) 999492999484

119910) ge [120572 120573] that is (119909 rarr 119910) 999492999484 119910 isin 119860(120572120573)

Similarly we canshow that119860

(120572120573)satisfies (SVF4) Hence119860

(120572120573)is a subpositive

implicative filter of 119871Conversely let 119905

119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) =

120572 and 1 minus 119891119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) = 120573 Thus

119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) ge [120572 120573] We get that

(119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909) isin 119860(120572120573)

So 119860(120572120573)

= 0Thus (119909 rarr 119910) 999492999484 119910 isin 119860

(120572120573) that is 119881

119860((119909 rarr 119910) 999492999484 119910) ge

[120572 120573] = 119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) Similarly we can

prove that 119860 satisfies (SVF4)

In the following theorems we obtain some characteriza-tions of subpositive implicative vague filter

Theorem 38 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF5) iminVA((y rarr z) 999492999484 (x rarr y))VA(x) le

VA(y) for any 119909 119910 119911 isin 119871(SVF6) iminVA((y 999492999484 z) rarr (x 999492999484 y))VA(x) le

VA(y) for any 119909 119910 119911 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871 ByProposition 2 part (3) we have imin119881

119860((119910 rarr 119911) 999492999484 (119909 rarr

119910)) 119881119860(119909) = imin119881

119860(119909 rarr ((119910 rarr 119911) 999492999484 119910)) 119881

119860(119909) le

119881119860((119910 rarr 119911) 999492999484 119910) By Proposition 2 part (7) (119910 rarr 119911) 999492999484

119910 le 119911 999492999484 119910 le (119910 rarr 119911) 999492999484 ((119911 999492999484 119910) rarr 119910) We get that119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119911 999492999484 119910) le 119881

119860((119910 rarr 119911) 999492999484 ((119911 999492999484

119910) rarr 119910)) le 119881119860((119911 999492999484 119910) rarr 119910) Hence imin119881

119860((119910 rarr

119911) 999492999484 119910) 119881119860(119911 999492999484 119910) le imin119881

119860((119911 999492999484 119910) rarr 119910) 119881

119860(119911 999492999484

119910) le 119881119860(119910) Thus it satisfies (SVF5) Similarly we can

prove (SVF6) Conversely let 119860 satisfy (SVF5) and (SVF6)119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) = imin119881

119860([((119909 rarr 119910) 999492999484

119910) rarr 1] 999492999484 [(119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909) rarr ((119909 rarr

119910) 999492999484 119910)]) 119881119860(1) le 119881

119860((119909 rarr 119910) 999492999484 119910) Similarly we

can prove that (SVF5) holds Hence 119860 is a vague subpositiveimplicative filter

Theorem 39 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF7) 119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) le

119881119860((119910 999492999484 119909) rarr 119909) for any 119909 119910 isin 119871

(SVF8) 119881119860((119909 999492999484 119910) rarr ((119910 999492999484 119909) rarr 119909)) le

119881119860((119910 rarr 119909) 999492999484 119909) for any 119909 119910 isin 119871

Proof Let (SVF7) and (SVF8) hold We will show that 119860

satisfies (SVF5) and (SVF6) We have imin119881119860((119910 rarr 119911) 999492999484

(119909 rarr 119910)) 119881119860(119909) = imin119881

119860(119909 rarr ((119910 rarr 119911) 999492999484

119910)) 119881119860(119909) le 119881

119860((119910 rarr 119911) 999492999484 119910) le 119881

119860((119910 rarr 119911) 999492999484

8 Journal of Discrete Mathematics

((119911 rarr 119910) 999492999484 119910)) le 119881119860((119911 999492999484 119910) rarr 119910)) Also we have

119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119911 999492999484 119910) Since 119860 is a vague

filter 119881119860((119910 rarr 119911) 999492999484 119910) le imin119881

119860(119911 999492999484 119910) 119881

119860((119911 999492999484

119910) rarr 119910) le 119881119860(119910) Hence 119860 satisfies (SVF5) Analogously

we can prove that119860 satisfies (SVF6) Hence119860 is a subpositiveimplicative vague filter of 119871 byTheorem 38 Conversely let 119860be a subpositive implicative vague filter of 119871 We have

[((119910 999492999484 119909) 997888rarr 119909) 997888rarr 1]

999492999484 [((119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909))

997888rarr ((119910 999492999484 119909) 997888rarr 119909) ]

= [(119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)]

997888rarr [((119910 997888rarr 119909) 999492999484 119909) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)]

= (119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)

(30)

We get that 119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) =

imin119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909))119881

119860([((119910 999492999484 119909) rarr

119909) rarr 1] 999492999484 [((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) rarr ((119910 999492999484

119909) rarr 119909)]) le 119881119860((119910 999492999484 119909) rarr 119909) by (SVF5) Similarly we

can prove that (SVF8) holds

Theorem 40 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF9) 119881119860((119909 rarr 119910) 999492999484 119909) le 119881

119860(119909) for any 119909 119910 isin 119871

(SVF10) 119881119860((119909 999492999484 119910) rarr 119909) le 119881

119860(119909) for any 119909 119910 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871Then 119860 satisfies (SVF5) and (SVF6)

119881119860((119909 997888rarr 119910) 999492999484 119909)

= imin 119881119860(1 997888rarr ((119909 997888rarr 119910) 999492999484 119909)) 119881

119860 (1)

= imin 119881119860((119909 997888rarr 119910) 999492999484 (1 997888rarr 119909)) 119881119860 (1)

le 119881119860 (119909)

(31)

Similarly we can prove that (SVF10) holds Conversely let119860 satisfy (SVF9) and (SVF10) We will show that 119860 satisfies(SVF5) and (SVF6) imin119881

119860((119910 rarr 119911) 999492999484 (119909 rarr

119910)) 119881119860(119909) = imin119881

119860((119909 rarr (119910 rarr 119911) 999492999484 119910)) 119881

119860(119909) le

119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119910) Hence 119860 satisfies (SVF5)

Similarly we can prove (SVF7) holds Therefore 119860 is asubpositive implicative vague filter of L byTheorem 38

Theorem 41 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF11) 119881119860(not119909 999492999484 119909) le 119881

119860(119909) for any 119909 isin 119871

(SVF12) 119881119860(sim 119909 rarr 119909) le 119881

119860(119909) for any 119909 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871Then (SVF11) and (SVF12) follow by (SVF9) and (SVF10)Conversely since 0 le 119910 then 119909 rarr 0 le 119909 rarr 119910 Hence

(119909 rarr 119910) 999492999484 119909 le not119909 999492999484 119909 by Proposition 2 part (6)Therefore 119881

119860((119909 rarr 119910) 999492999484 119909) le 119881

119860(not119909 999492999484 119909) le 119881

119860(119909)

Similarly we can prove that (SVF10) holds Hence 119860 is asubpositive implicative vague filter of 119871 byTheorem 40

The extension theorem of subpositive implicative vaguefilters is obtained from the following theorem

Theorem 42 Let 119860 and 119861 be two vague filters of a residuatedlattice 119871 which satisfies 119860 le 119861 and 119881

119860(1) = 119881

119861(1) If 119860 is a

subpositive implicative vague filter of L so 119861 is

Proof Suppose that 119906 = (119909 rarr 119910) 999492999484 119909 Then

1 = 119906 997888rarr 119906 = 119906 997888rarr ((119909 997888rarr 119910) 999492999484 119909)

= (119909 997888rarr 119910) 999492999484 (119906 997888rarr 119909) = 1

(32)

We have (119909 rarr 119910) 999492999484 (119906 rarr 119909) le ((119906 rarr 119909) rarr 119910) 999492999484

(119906 rarr 119909) Therefore 119881119860(((119906 rarr 119909) rarr 119910) 999492999484 (119906 rarr 119909)) =

119881119860(1) By (SVF10) 119881

119860(1) = 119881

119860(((119906 rarr 119909) rarr 119910) 999492999484 (119906 rarr

119909)) le 119881119860(119906 rarr 119909) Hence 119881

119860(119906 rarr 119909) = 119881

119860(1) = 119881

119861(1)

Since 119860 le 119861 then 119881119861(119906 rarr 119909) ge 119881

119860(119906 rarr 119909) = 119881

119861(1)

We get that 119881119861((119909 rarr 119910) 999492999484 119909) = 119881

119861(119906) = imin119881

119861(119906 rarr

119909) 119881119861(119906) le 119881

119861(119909) Similarly we can show that 119866 satisfies

(SVF11) Hence 119866 is a vague subpositive implicative filter of119871 by Theorem 40

Definition 43 Let 119860 be a vague filter of a residuated lattice 119871119860 is called a vague Boolean filter of 119871 if

(BVF1) 119881119860(119909 or not119909) = 119881

119860(1) for all 119909 isin 119871

(BVF2) 119881119860(119909or sim 119909) = 119881

119860(1) for all 119909 isin 119871

Similar to Theorem 40 we have the following theoremwith respect to Boolean vague filters

Theorem 44 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a Boolean vague filter of 119871 if and only if for all120572 120573 isin [0 1] the set 119860

(120572120573)is either empty or a Boolean filter of

119871 where 120572 le 120573

Theorem 45 Let 119860 be a vague filter of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and if onlyif 119860 is a Boolean vague filter

Proof Let119860 be a subpositive implicative vague filterWe have

119881119860 (1) = 119881

119860 (not (119909 or not119909) 999492999484 (119909 or not119909))

= 119881119860 ((not119909 and notnot119909) 999492999484 (119909 or not119909)) le 119881

119860 (119909 or not119909)

(33)

Hence 119881119860(119909 or not119909) = 119881

119860(1) Similarly we can prove that 119860

satisfies (BVF2) Hence 119860 is a Boolean vague filterConversely let119881

119860(119909ornot119909) = 119881

119860(1)We have119881

119860(119909ornot119909) =

119881119860((119909 999492999484 119909) or (not119909 999492999484 119909))119881

119860((119909 or not119909) 999492999484 119909) = imin119881

119860((119909 or

not119909) 999492999484 119909) 119881119860(1) = imin119881

119860((119909 or not119909) 999492999484 119909) 119881

119860(119909 or not119909) le

119881119860(119909) Similarly we can show that119860 satisfies (SVF12) Hence

119860 is a subpositive implicative vague filter byTheorem 41

Journal of Discrete Mathematics 9

6 Conclusion

In this paper we introduced the notions of vague filterssubpositive implicative vague filters and Boolean vaguefilters of a residuated lattice and investigated some relatedproperties We studied the relationship between filters andvague filters of a residuated lattice and showed that vaguefilters of a residuated lattice is a generalization of fuzzy filtersWe proved that the set of all vague filters of a residuated latticeforms a complete lattice and obtained some characterizationsof subpositive implicative vague filterThe extension theoremof subpositive implicative vague filters was obtained Finallyit was proved that subpositive implicative vague filters areequivalent to Boolean vague filters

Conflict of Interests

There is no conflict of interests regarding the publication ofthis paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993

[3] Y B Jun and C H Park ldquoIdeals of pseudo MV-algebras basedon vague set theoryrdquo Iranian Journal of Fuzzy Systems vol 6 no7 pp 31ndash47 2009

[4] A Borumand Saeid ldquoVague BCKBCI-algebrasrdquo OpusculaMathematica vol 29 no 2 pp 177ndash186 2009

[5] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939

[6] T S Blyth and M F Janowitz Residuation Theory PergamonPress 1972

[7] T Kowalski and H Ono Residuated Lattices An AlgebraicGlimpse at Logic without Contraction Japan Advanced Instituteof Science and Technology March 2001

[8] D PiciuAlgebras of Fuzzy Logic Editura Universitaria CraiovaCraiova Romania 2007

[9] E TurunenMathematics Behind Fuzzy Logic Physica 1999[10] P Hajek Metamathematics of Fuzzy Logic vol 4 of Trends

in LogicmdashStudia Logica Library Kluwer Academic DordrechtThe Netherlands 1998

[11] Sh Ghorbani ldquoSub positive implicative filters of residuatedlatticesrdquo World Applied Science Journal vol 12 no 5 pp 586ndash590 2011

[12] S Ghorbani and A Hasankhani ldquoFuzzy convex subalgebrasof commutative residuated latticesrdquo Iranian Journal of FuzzySystems vol 7 no 2 pp 41ndash54 159 2010

[13] R Biswas ldquoVague groupsrdquo International Journal of Computa-tional Cognition vol 4 pp 20ndash23 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Discrete Mathematics

Definition 18 A vague set 119860 of a residuated lattice 119871 is calleda vague lattice filter of 119871 if it satisfies (VF1) and

(VLF2) imin119881119860(119909) 119881119860(119910) le 119881

119860(119909 and 119910) for all

119909 119910 isin 119871

Proposition 19 Let 119860 be a vague filter of a residuated lattice119871 then 119860 is a vague lattice filter

In the following example we will show that the converseof Proposition 19 may not be true in general

Example 20 Let 0 119886 119887 119888 1 such that 0 lt 119886 lt 119887 119888 lt 1where 119887 and 119888 are incomparable Define the operations lowast rarr and 999492999484 as follows

lowast 0 119886 119887 119888 1

0 0 0 0 0 0

119886 0 0 0 119886 119886

119887 0 119886 119887 119886 119887

119888 0 0 0 119888 119888

1 0 119886 119887 119888 1

997888rarr 0 119886 119887 119888 1

0 1 1 1 1 1

119886 119888 1 1 1 1

119887 119888 119888 1 119888 1

119888 0 119887 119887 1 1

1 0 119886 119887 119888 1

999492999484 0 119886 119887 119888 1

0 1 1 1 1 1

119886 119887 1 1 1 1

119887 0 119888 1 119888 1

119888 119887 119887 119887 1 1

1 0 119886 119887 119888 1

(9)

Then 119871 is a residuated lattice Let 119860 be the vague set definedas follows

119860 = ⟨0 [01 02]⟩ ⟨119886 [03 04]⟩ ⟨119887 [03 04]⟩

⟨119888 [03 08]⟩ ⟨0 [05 09]⟩

(10)

It is routine to verify that119860 is a vague lattice filter but it is nota vague filter of 119871

Let 119861 be the vague set defined as follows

119861 = ⟨0 [02 05]⟩ ⟨119886 [02 05]⟩ ⟨119887 [02 05]⟩

⟨119888 [04 07]⟩ ⟨0 [07 07]⟩

(11)

Then 119861 is a vague filter of 119871

In the following theorems we will study the relationshipbetween filters and vague filters of a residuated lattice

Theorem 21 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter of 119871 if and only if for all 120572 120573 isin [0 1]the set 119860

(120572120573)is either empty or a filter of 119871 where 120572 le 120573

Proof Suppose that 119860 is a vague filter of 119871 and 120572 120573 isin [0 1]Let 119860

(120572120573)= 0 We will show that 119860

(120572120573)is a filter of 119871

(1) Suppose that 119909 le 119910 and 119909 isin 119860(120572120573)

By (VF1)119881119860(119910) ge

119881119860(119909) ge [120572 120573] Therefor 119910 isin 119860

(120572120573)

(2) Suppose that 119909 119910 isin 119860(120572120573)

Then 119881119860(119909) 119881119860(119910) ge

[120572 120573] By (VF2) we have 119881119860(119909 lowast 119910) ge

imin119881119860(119909) 119881119860(119910) ge [120572 120573] Hence 119909 lowast 119910 isin 119860

(120572120573)

and then 119860(120572120573)

is a filter of 119871

Conversely suppose that for all 120572 120573 isin [0 1] the sets 119860(120572120573)

is either empty or a filter of 119871 Let 119905119860(119909) = 120572

1 119905119860(119910) = 120572

2

1 minus 119891119860(119909) = 120573

1 and 1 minus 119891

119860(119910) = 120573

2 Put 120572 = min120572

1 1205722

and 120573 = min1 minus 1205731 1 minus 120573

2 Then 119905

119860(119909) 119905119860(119910) ge 120572 and 1 minus

119891119860(119909) 1 minus 119891

119860(119910) ge 120573 Hence 119881

119860(119910) 119881119860(119909) ge [120572 120573] that is

119909 119910 isin 119860(120572120573)

So 119860(120572120573)

= 0 and by assumption 119860(120572120573)

is afilter of 119871 We obtain that 119909 lowast 119910 isin 119860

(120572120573) that is

119881119860(119909 lowast 119910) ge [120572 120573]

= [min 1205721 1205722 min 1 minus 120573

1 1 minus 120573

2]

= imin [119905119860 (119909) 1 minus 119891

119860 (119909)] [119905119860 (119910) 1 minus 119891119860(119910)]

= imin 119881119860 (119909) 119881119860 (119910)

(12)

Hence (VF2) holdsSuppose that 119909 le 119910 119905

119860(119909) = 120572 and 1 minus 119891

119860(119909) = 120573 Hence

119909 isin 119860(120572120573)

By assumption 119860(120572120573)

is a filter We get that 119910 isin

119860(120572120573)

that is 119881119860(119910) ge [120572 120573] = [119905

119860(119909) 1 minus 119891

119860(119909)] = 119881

119860(119909)

Hence 119860 is a vague filter of 119871

The filters like119860(120572120573)

are also called vague-cuts filters of 119871

Corollary 22 Let 119860 be a vague filter of a residuated lattice 119871Then for all 120572 isin [0 1] the set 119860

120572is either empty or a filter of

119871

Corollary 23 Any filter 119865 of a residuated lattice 119871 is a vague-cut filter of some vague filter of 119871

Proof For all 119909 isin 119871 define

1 minus 119891119860 (119909) = 119905

119860 (119909) = 1 if 119909 isin 119865

120572 otherwise(13)

Hence we have 119881119860(119909) = [1 1] if 119909 isin 119865 and 119881

119860(119909) = [120572 120572]

otherwise where 120572 isin (0 1) It is clear that 119865 = 119860(11)

Let119909 119910 isin 119871 If 119909 119910 isin 119865 then 119909 lowast 119910 isin 119865 We have 119881

119860(119909 lowast 119910) =

[1 1] = imin119881119860(119909) 119881119860(119910) If one of 119909 or 119910 does not belong

to 119865 then one of119881119860(119909) and119881

119860(119910) is equal to [120572 120572]Therefore

119881119860(119909lowast119910) ge [120572 120572] = imin119881

119860(119909) 119881119860(119910) Suppose that 119909 le 119910

If 119909 isin 119865 then 119910 isin 119865 Hence 119881119860(119909) = 119881

119860(119910) If 119909 notin 119865 then

119881119860(119909) = [120572 120572] We have 119881

119860(119909) le 119881

119860(119910) Hence 119860 is a vague

filter of 119871

In the following proposition we will show that vaguefilters of a residuated lattice are a generalization of fuzzyfilters

Proposition 24 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a vague filter of 119871 if and only if the fuzzy sets 119905

119860and

1 minus 119891119860are fuzzy filters in 119871

Journal of Discrete Mathematics 5

4 Lattice of Vague Filters

Let 119860 and 119861 be vague sets of a residuated lattice 119871 If 119881119860(119909) le

119881119861(119909) for all 119909 isin 119871 then we say 119861 containing119860 and denote it

by 119860 ⪯ 119861The unite vague set of a residuated lattice 119871 is a vague

filter of 119871 containing any vague filter of 119871 For a vague setwe can define the intersection of two vague sets 119860 and 119861 of aresiduated lattice 119871 by 119881

119860cap119861(119909) = imin119881

119860(119909) 119881119861(119909) for all

119909 isin 119871 It is easy to prove that the intersection of any family ofvague filters of 119871 is a vague filter of 119871 Hence we can define avague filter generated by a vague set as follows

Definition 25 Let 119860 be a vague set of a residuated lattice 119871A vague filter 119861 is called a vague filter generated by 119860 if itsatisfies

(i) 119860 ⪯ 119861(ii) 119860 ⪯ 119862 that implies 119861 ⪯ 119862 for all vague filters 119862 of 119871

The vague filter generated by 119860 is denoted by ⟨119860⟩

Example 26 Consider the vague lattice filter119860 in Example 20which is not a vague filter of 119871Then the vague filter generatedby 119860 is

⟨119860⟩ = ⟨0 [03 04]⟩ ⟨119886 [03 04]⟩

⟨119887 [03 04]⟩ ⟨119888 [03 08]⟩ ⟨0 [05 09]⟩

(14)

Theorem 27 Let 119860 be a vague set of a residuated lattice 119871Then the vague value of 119909 in ⟨119860⟩ is

119881⟨119860⟩ (119909) = isup imin VA (a1) VA (a2) VA (an)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

(15)

for all 119909 isin 119871

Proof First we will prove that ⟨119860⟩ is a vague set of 119871

119905⟨119860⟩ (119909)

= sup min 119905119860(1198861) 119905119860(1198862) 119905

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

le sup min 1 minus 119891119860(1198861) 1 minus 119891

119860(1198862) 1 minus 119891

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

= sup 1 minus max 119891119860(1198861) 119891119860(1198862) 119891

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

= 1 minus inf max 119891119860(1198861) 119891119860(1198862) 119891

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

= 1 minus 119891⟨119860⟩ (119909)

(16)

Now we will show that ⟨119860⟩ is a vague filter of 119871(VF1) Suppose that 119909 119910 isin 119871 is arbitrary For every 120576 gt 0

we can take 1198861 119886

119899 1198871 119887

119898isin 119871 such that119909 ge 119886

1lowastsdot sdot sdotlowast119886

119899

119910 ge 1198871lowast sdot sdot sdot lowast 119887

119898 and

119905⟨119860⟩ (119909) minus 120576 lt min 119905

119860(1198861) 119905119860(1198862) 119905

119860(119886119899)

119905⟨119860⟩

(119910) minus 120576 lt min 119905119860(1198871) 119905119860(1198872) 119905

119860(119887119898)

(17)

By Proposition 2 part (4) we have 119909 lowast 119910 ge 1198861lowast sdot sdot sdot lowast 119886

119899lowast 1198871lowast

sdot sdot sdot lowast 119887119898and then

119905⟨119860⟩

(119909 lowast 119910)

ge min 119905119860(1198861) 119905

119860(119886119899) 119905119860(1198871) 119905

119860(119887119898)

= min min 119905119860(1198861) 119905

119860(119886119899)

min 119905119860(1198871) 119905

119860(119887119898)

gt min (119905⟨119860⟩ (119909) minus 120576) (119905

⟨119860⟩(119910) minus 120576)

= min 119905⟨119860⟩ (119909) 119905⟨119860⟩ (119910) minus 120576

(18)

Since 120576 gt 0 is arbitrary we obtain that 119905⟨119860⟩

(119909 lowast 119910) ge

min119905⟨119860⟩

(119909) 119905⟨119860⟩

(119910)Similarly we can show that 1 minus 119891

⟨119860⟩(119909 lowast 119910) ge min1 minus

119891⟨119860⟩

(119909) 1 minus119891⟨119860⟩

(119910) Hence119881119860(119909lowast119910) ge imin119881

119860(119909) 119881119860(119910)

for all 119909 119910 isin 119871(VF2) Suppose that 119909 le 119910where 119909 119910 isin 119871 Let 119886

1 119886

119899isin

119871 such that 1198861lowast sdot sdot sdot lowast 119886

119899le 119909 Then 119886

1lowast sdot sdot sdot lowast 119886

119899le 119910 And we

get that

119881119860(119910) ge imin 119881

119860(1198861) 119881119860(1198862) 119881

119860(119886119899) (19)

Since 1198861 119886

119899isin 119871 where 119909 ge 119886

1lowast sdot sdot sdot lowast 119886

119899is arbitrary then

119881⟨119860⟩

(119910) ge isup imin 119881119860(1198861) 119881119860(1198862) 119881

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

= 119881⟨119860⟩ (119909)

(20)

Therefore ⟨119860⟩ is a vague filter of 119871 by Theorem 16 It is clearthat119860 ⪯ ⟨119860⟩ Now let119862 be a vague filter of119871 such that119860 ⪯ 119862Then we have

119881⟨119860⟩ (119909)

= isup imin 119881119860(1198861) 119881

119860(119886119899) 119909 ge 119886

1lowast sdot sdot sdot lowast 119886

119899

le isup imin 119881119862(1198861) 119881

119862(119886119899) 119909 ge 119886

1lowast sdot sdot sdot lowast 119886

119899

= 119881119862 (119909)

(21)

for all 119909 isin 119871 Hence ⟨119860⟩ ⪯ 119862 and then ⟨119860⟩ is a vague filtergenerated by 119860

We denote the set of all vague filters of a residuated lattice119871 by VF(119871)

6 Journal of Discrete Mathematics

Corollary 28 (VF(119871) cap or) is a complete lattice

Proof Clearly if 119860119894119894isinΓ

is a family of vague filters of aresiduated lattice 119871 then the infimumof this family is⋂

119894isinΓ119860119894

and the supremum is ⋁119894isinΓ

119860119894= ⟨⋃119894isinΓ

119860119894⟩ It is easy to prove

that (IFF(119871) cap or) is a complete lattice

Let 119860 and 119861 be vague sets of a residuated lattice 119871 Then119860 lowast 119861 is defined as follows

119881(119860lowast119861) (119911) = isup imin 119881

119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910 (22)

for all 119911 isin 119871

Theorem 29 Let 119860 and 119861 be vague sets of a residuated lattice119871 Then 119860 lowast 119861 is a vague set of 119871

Proof Since 119860 and 119861 are vague sets of 119871 we have 119905119860(119909) le

1 minus 119891119860(119909) and 119905

119861(119910) le 1 minus 119891

119861(119910) for all 119909 119910 isin 119871 Thus

119905119860lowast119861 (119911) = sup min 119905

119860 (119909) 119905119860 (119910) 119911 ge 119909 lowast 119910

le sup min 1 minus 119891119860 (119909) 1 minus 119891

119860(119910) 119911 ge 119909 lowast 119910

le 1 minus 119891119860lowast119861 (119911)

(23)

Hence 119905119860lowast119861

(119911) + 119891119860lowast119861

(119911) le 1 and then 119860 lowast 119861 is a vague set of119871

Theorem30 Let119860 and119861 be vague filters of a residuated lattice119871 Then 119860 lowast 119861 is a vague filter of 119871

Proof ByTheorem 29119860lowast119861 is a vague set of 119871 We will provethat 119860 lowast 119861 is a vague filter of 119871

(VF2) Let 1199111

le 1199112and 119909 lowast 119910 le 119911

1 We get that

imin119881119860(119909) 119881119861(119910) le 119881

119860lowast119861(1199112) Since 119909 lowast 119910 le 119911

1is arbitrary

we have

119881119860lowast119861

(1199111) = isup imin 119881

119860 (119909) 119881119861 (119910) 1199111ge 119909 lowast 119910

le 119881119860lowast119861

(1199112)

(24)

(VF1) Let 119909 119910 isin 119871 Then

imin 119881119860lowast119861 (119909) 119881119860lowast119861 (119910)

= imin isup imin 119881119860 (119886) 119881119861 (119887) 119909 ge 119886 lowast 119887

isup imin 119881119860 (119888) 119881119861 (119889) 119910 ge 119888 lowast 119889

= isup isup imin 119881119860 (119886 lowast 119888) 119881119861 (119888 lowast 119889)

119909 ge 119886 lowast 119887 119910 ge 119888 lowast 119889

le isup imin 119881119860 (119886 lowast 119888) 119881119861 (119888 lowast 119889)

119909 lowast 119910 ge (119886 lowast 119887) lowast (119888 lowast 119889)

le isup imin 119881119860 (119890) 119881119861 (119891) 119909 lowast 119910 ge 119890 lowast 119891

= 119881119860lowast119861

(119909 lowast 119910)

(25)

By Definition 13 119860 lowast 119861 is a vague filter of 119871

Theorem31 Let119860 and119861 be vague filters of a residuated lattice119871 such that119881

119860(1) = 119881

119861(1) Then119860lowast119861 is the least upper bound

of 119860 and 119861

Proof By Definition 13 we have 119881119860(1) = 119881

119861(1) ge 119881

119860(119910) for

all 119910 isin 119871 Let 119911 isin 119871 Since 119911 = 1 lowast 119911 we get that

119881119860lowast119861 (119911) = isup imin 119881

119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910

ge imin 119881119860 (119911) 119881119861 (1) ge 119881

119860 (119911)

(26)

Hence 119860 ⪯ 119860 lowast 119861 Similarly we can show that 119861 ⪯ 119860 lowast 119861Suppose that 119862 is the vague filter of 119871 containing 119860 and 119861Then

119881119860lowast119861 (119911)

= isup imin 119881119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910

le isup imin 119881119862 (119909) 119881119862 (119910) 119911 ge 119909 lowast 119910

le isup imin 119881119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 = 120583

119862 (119911)

(27)

Hence 119860lowast 119861 ⪯ 119862 that is 119860lowast 119861 is the least upper bound of 119860and 119861

Lemma 32 Let119862 be a vague filter of a residuated lattice 119871 and119911 isin 119871 Then

(1) 119881119862(119911) = isupVC(x lowast y) z ge x lowast y for all 119909 119910 119911 isin 119871

(2) 119881119862(119909 lowast 119910) = iminVC(x)VC(y) for 119909 119910 isin 119871

Proof (1) Since 119909 lowast 119910 le 119911 then 119881119862(119909 lowast 119910) le 119881

119862(119911) by (VF2)

Hence 119881119862(119911) ge isup119881

119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 Also we have

1 lowast 119911 = 119911 We obtain that

119881119862 (119911) = 119881

119862 (1 lowast 119911) le isup 119881119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 (28)

(2) It follows from Proposition 2 part (2) andDefinition 13

Let VFD(119871) be the set of all vague filters119860 and119861 of 119871 suchthat 119881

119860(1) = 119881

119861(1)

Theorem 33 (119881119865119863(119871) cap lowast) is a distributive lattice

Journal of Discrete Mathematics 7

Proof Let 119860 119861 119862 isin VFD(119871) be such that 119860 ⪯ 119862 We willshow that (119860 lowast 119861) cap 119862 = (119860 lowast 119862) cap (119861 lowast 119862) By Lemma 32

119881(119860lowast119861)cap119862 (119911)

= imin (119881119860lowast119861

) (119911) 119881119862 (119911)

= imin isup imin 119881119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910 119881

119862 (119911)

= isup imin 119881119860 (119909) 119881119861 (119910) 119881

119862 (119911) 119911 ge 119909 lowast 119910

= isup imin 119881119860 (119909) 119881119861 (119910) 119881

119862(119909 lowast 119910) 119911 ge 119909 lowast 119910

= isup imin 119881119860 (119909) 119881119861 (119910) 119881

119862 (119909) 119881119862 (119910) 119911 ge 119909 lowast 119910

= isup imin 119881119860 (119909) 119881119862 (119909)

imin 119881119861(119910) 119881

119862(119910) 119911 ge 119909 lowast 119910

= isup imin 119881119860cap119862 (119909) 119881119861cap119862 (119910) 119911 ge 119909 lowast 119910

= 119881(119860cap119862)lowast(119861cap119862) (119911)

(29)

5 Subpositive Implicative Vague Filters

Definition 34 Let 119860 be a vague subset of a residuated lattice119871 119860 is called a subpositive implicative vague filter of 119871 if

(VF6) 119881119860(119909) le 119881

119860(1) for all 119909 isin 119871

(SVF1) imin119881119860(((119909 rarr 119910) lowast 119911) 999492999484 ((119910 999492999484 119909) rarr

119909)) 119881119860(119911) le 119881

119860((119909 rarr 119910) 999492999484 119910) for any

119909 119910 119911 isin 119871(SVF2) imin119881

119860((119911 lowast (119909 999492999484 119910)) rarr ((119910 rarr 119909) 999492999484

119909)) 119881119860(119911) le 119881

119860((119909 999492999484 119910) rarr 119910) for any

119909 119910 119911 isin 119871

Proposition 35 Let 119860 be a vague filter of a residuated lattice119871 Then 119860 is a subpositive implicative vague filter of L if andonly if it satisfies

(SVF3) 119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) le

119881119860((119909 rarr 119910) 999492999484 119910) for any 119909 119910 isin 119871

(SVF4) 119881119860((119909 999492999484 119910) rarr ((119910 rarr 119909) 999492999484 119909)) le

119881119860((119909 999492999484 119910) rarr 119910) for any 119909 119910 isin 119871

Proposition 36 Let119860 be a subpositive implicative vague filterof a residuated lattice 119871 Then 119860 is a vague filter of 119871

Proof We have imin119881119860(119909 999492999484 119910) 119881

119860(119909) = imin119881

119860((119910 rarr

119910) lowast 119909) 999492999484 ((119910 999492999484 119910) rarr 119910) 119881119860(119909) le 119881

119860((119910 rarr 119910) 999492999484

119910) = 119881119860(119910) By (VS1) andTheorem 17 119860 is a vague filter of a

residuated lattice 119871

The relation between subpositive implicative vague filterand its level subset is as follows

Theorem 37 Let 119860 be a vague set of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and only if

for all120572 120573 isin [0 1] the set119860(120572120573)

is either empty or a subpositiveimplicative filter of 119871 where 120572 le 120573

Proof Suppose that119860 is a subpositive implicative filter vagueset of 119871 and 120572 120573 isin [0 1] Let 119860

(120572120573)= 0 Since 119860 is a vague

filter then119860(120572120573)

is a filter of 119871 byTheorem 21 Now let (119909 rarr

119910) 999492999484 ((119910 999492999484 119909) rarr 119909) isin 119860(120572120573)

Then 119881119860((119909 rarr 119910) 999492999484

((119910 999492999484 119909) rarr 119909)) ge [120572 120573] By (SVF3) 119881119860((119909 rarr 119910) 999492999484

119910) ge [120572 120573] that is (119909 rarr 119910) 999492999484 119910 isin 119860(120572120573)

Similarly we canshow that119860

(120572120573)satisfies (SVF4) Hence119860

(120572120573)is a subpositive

implicative filter of 119871Conversely let 119905

119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) =

120572 and 1 minus 119891119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) = 120573 Thus

119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) ge [120572 120573] We get that

(119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909) isin 119860(120572120573)

So 119860(120572120573)

= 0Thus (119909 rarr 119910) 999492999484 119910 isin 119860

(120572120573) that is 119881

119860((119909 rarr 119910) 999492999484 119910) ge

[120572 120573] = 119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) Similarly we can

prove that 119860 satisfies (SVF4)

In the following theorems we obtain some characteriza-tions of subpositive implicative vague filter

Theorem 38 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF5) iminVA((y rarr z) 999492999484 (x rarr y))VA(x) le

VA(y) for any 119909 119910 119911 isin 119871(SVF6) iminVA((y 999492999484 z) rarr (x 999492999484 y))VA(x) le

VA(y) for any 119909 119910 119911 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871 ByProposition 2 part (3) we have imin119881

119860((119910 rarr 119911) 999492999484 (119909 rarr

119910)) 119881119860(119909) = imin119881

119860(119909 rarr ((119910 rarr 119911) 999492999484 119910)) 119881

119860(119909) le

119881119860((119910 rarr 119911) 999492999484 119910) By Proposition 2 part (7) (119910 rarr 119911) 999492999484

119910 le 119911 999492999484 119910 le (119910 rarr 119911) 999492999484 ((119911 999492999484 119910) rarr 119910) We get that119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119911 999492999484 119910) le 119881

119860((119910 rarr 119911) 999492999484 ((119911 999492999484

119910) rarr 119910)) le 119881119860((119911 999492999484 119910) rarr 119910) Hence imin119881

119860((119910 rarr

119911) 999492999484 119910) 119881119860(119911 999492999484 119910) le imin119881

119860((119911 999492999484 119910) rarr 119910) 119881

119860(119911 999492999484

119910) le 119881119860(119910) Thus it satisfies (SVF5) Similarly we can

prove (SVF6) Conversely let 119860 satisfy (SVF5) and (SVF6)119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) = imin119881

119860([((119909 rarr 119910) 999492999484

119910) rarr 1] 999492999484 [(119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909) rarr ((119909 rarr

119910) 999492999484 119910)]) 119881119860(1) le 119881

119860((119909 rarr 119910) 999492999484 119910) Similarly we

can prove that (SVF5) holds Hence 119860 is a vague subpositiveimplicative filter

Theorem 39 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF7) 119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) le

119881119860((119910 999492999484 119909) rarr 119909) for any 119909 119910 isin 119871

(SVF8) 119881119860((119909 999492999484 119910) rarr ((119910 999492999484 119909) rarr 119909)) le

119881119860((119910 rarr 119909) 999492999484 119909) for any 119909 119910 isin 119871

Proof Let (SVF7) and (SVF8) hold We will show that 119860

satisfies (SVF5) and (SVF6) We have imin119881119860((119910 rarr 119911) 999492999484

(119909 rarr 119910)) 119881119860(119909) = imin119881

119860(119909 rarr ((119910 rarr 119911) 999492999484

119910)) 119881119860(119909) le 119881

119860((119910 rarr 119911) 999492999484 119910) le 119881

119860((119910 rarr 119911) 999492999484

8 Journal of Discrete Mathematics

((119911 rarr 119910) 999492999484 119910)) le 119881119860((119911 999492999484 119910) rarr 119910)) Also we have

119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119911 999492999484 119910) Since 119860 is a vague

filter 119881119860((119910 rarr 119911) 999492999484 119910) le imin119881

119860(119911 999492999484 119910) 119881

119860((119911 999492999484

119910) rarr 119910) le 119881119860(119910) Hence 119860 satisfies (SVF5) Analogously

we can prove that119860 satisfies (SVF6) Hence119860 is a subpositiveimplicative vague filter of 119871 byTheorem 38 Conversely let 119860be a subpositive implicative vague filter of 119871 We have

[((119910 999492999484 119909) 997888rarr 119909) 997888rarr 1]

999492999484 [((119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909))

997888rarr ((119910 999492999484 119909) 997888rarr 119909) ]

= [(119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)]

997888rarr [((119910 997888rarr 119909) 999492999484 119909) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)]

= (119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)

(30)

We get that 119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) =

imin119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909))119881

119860([((119910 999492999484 119909) rarr

119909) rarr 1] 999492999484 [((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) rarr ((119910 999492999484

119909) rarr 119909)]) le 119881119860((119910 999492999484 119909) rarr 119909) by (SVF5) Similarly we

can prove that (SVF8) holds

Theorem 40 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF9) 119881119860((119909 rarr 119910) 999492999484 119909) le 119881

119860(119909) for any 119909 119910 isin 119871

(SVF10) 119881119860((119909 999492999484 119910) rarr 119909) le 119881

119860(119909) for any 119909 119910 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871Then 119860 satisfies (SVF5) and (SVF6)

119881119860((119909 997888rarr 119910) 999492999484 119909)

= imin 119881119860(1 997888rarr ((119909 997888rarr 119910) 999492999484 119909)) 119881

119860 (1)

= imin 119881119860((119909 997888rarr 119910) 999492999484 (1 997888rarr 119909)) 119881119860 (1)

le 119881119860 (119909)

(31)

Similarly we can prove that (SVF10) holds Conversely let119860 satisfy (SVF9) and (SVF10) We will show that 119860 satisfies(SVF5) and (SVF6) imin119881

119860((119910 rarr 119911) 999492999484 (119909 rarr

119910)) 119881119860(119909) = imin119881

119860((119909 rarr (119910 rarr 119911) 999492999484 119910)) 119881

119860(119909) le

119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119910) Hence 119860 satisfies (SVF5)

Similarly we can prove (SVF7) holds Therefore 119860 is asubpositive implicative vague filter of L byTheorem 38

Theorem 41 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF11) 119881119860(not119909 999492999484 119909) le 119881

119860(119909) for any 119909 isin 119871

(SVF12) 119881119860(sim 119909 rarr 119909) le 119881

119860(119909) for any 119909 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871Then (SVF11) and (SVF12) follow by (SVF9) and (SVF10)Conversely since 0 le 119910 then 119909 rarr 0 le 119909 rarr 119910 Hence

(119909 rarr 119910) 999492999484 119909 le not119909 999492999484 119909 by Proposition 2 part (6)Therefore 119881

119860((119909 rarr 119910) 999492999484 119909) le 119881

119860(not119909 999492999484 119909) le 119881

119860(119909)

Similarly we can prove that (SVF10) holds Hence 119860 is asubpositive implicative vague filter of 119871 byTheorem 40

The extension theorem of subpositive implicative vaguefilters is obtained from the following theorem

Theorem 42 Let 119860 and 119861 be two vague filters of a residuatedlattice 119871 which satisfies 119860 le 119861 and 119881

119860(1) = 119881

119861(1) If 119860 is a

subpositive implicative vague filter of L so 119861 is

Proof Suppose that 119906 = (119909 rarr 119910) 999492999484 119909 Then

1 = 119906 997888rarr 119906 = 119906 997888rarr ((119909 997888rarr 119910) 999492999484 119909)

= (119909 997888rarr 119910) 999492999484 (119906 997888rarr 119909) = 1

(32)

We have (119909 rarr 119910) 999492999484 (119906 rarr 119909) le ((119906 rarr 119909) rarr 119910) 999492999484

(119906 rarr 119909) Therefore 119881119860(((119906 rarr 119909) rarr 119910) 999492999484 (119906 rarr 119909)) =

119881119860(1) By (SVF10) 119881

119860(1) = 119881

119860(((119906 rarr 119909) rarr 119910) 999492999484 (119906 rarr

119909)) le 119881119860(119906 rarr 119909) Hence 119881

119860(119906 rarr 119909) = 119881

119860(1) = 119881

119861(1)

Since 119860 le 119861 then 119881119861(119906 rarr 119909) ge 119881

119860(119906 rarr 119909) = 119881

119861(1)

We get that 119881119861((119909 rarr 119910) 999492999484 119909) = 119881

119861(119906) = imin119881

119861(119906 rarr

119909) 119881119861(119906) le 119881

119861(119909) Similarly we can show that 119866 satisfies

(SVF11) Hence 119866 is a vague subpositive implicative filter of119871 by Theorem 40

Definition 43 Let 119860 be a vague filter of a residuated lattice 119871119860 is called a vague Boolean filter of 119871 if

(BVF1) 119881119860(119909 or not119909) = 119881

119860(1) for all 119909 isin 119871

(BVF2) 119881119860(119909or sim 119909) = 119881

119860(1) for all 119909 isin 119871

Similar to Theorem 40 we have the following theoremwith respect to Boolean vague filters

Theorem 44 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a Boolean vague filter of 119871 if and only if for all120572 120573 isin [0 1] the set 119860

(120572120573)is either empty or a Boolean filter of

119871 where 120572 le 120573

Theorem 45 Let 119860 be a vague filter of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and if onlyif 119860 is a Boolean vague filter

Proof Let119860 be a subpositive implicative vague filterWe have

119881119860 (1) = 119881

119860 (not (119909 or not119909) 999492999484 (119909 or not119909))

= 119881119860 ((not119909 and notnot119909) 999492999484 (119909 or not119909)) le 119881

119860 (119909 or not119909)

(33)

Hence 119881119860(119909 or not119909) = 119881

119860(1) Similarly we can prove that 119860

satisfies (BVF2) Hence 119860 is a Boolean vague filterConversely let119881

119860(119909ornot119909) = 119881

119860(1)We have119881

119860(119909ornot119909) =

119881119860((119909 999492999484 119909) or (not119909 999492999484 119909))119881

119860((119909 or not119909) 999492999484 119909) = imin119881

119860((119909 or

not119909) 999492999484 119909) 119881119860(1) = imin119881

119860((119909 or not119909) 999492999484 119909) 119881

119860(119909 or not119909) le

119881119860(119909) Similarly we can show that119860 satisfies (SVF12) Hence

119860 is a subpositive implicative vague filter byTheorem 41

Journal of Discrete Mathematics 9

6 Conclusion

In this paper we introduced the notions of vague filterssubpositive implicative vague filters and Boolean vaguefilters of a residuated lattice and investigated some relatedproperties We studied the relationship between filters andvague filters of a residuated lattice and showed that vaguefilters of a residuated lattice is a generalization of fuzzy filtersWe proved that the set of all vague filters of a residuated latticeforms a complete lattice and obtained some characterizationsof subpositive implicative vague filterThe extension theoremof subpositive implicative vague filters was obtained Finallyit was proved that subpositive implicative vague filters areequivalent to Boolean vague filters

Conflict of Interests

There is no conflict of interests regarding the publication ofthis paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993

[3] Y B Jun and C H Park ldquoIdeals of pseudo MV-algebras basedon vague set theoryrdquo Iranian Journal of Fuzzy Systems vol 6 no7 pp 31ndash47 2009

[4] A Borumand Saeid ldquoVague BCKBCI-algebrasrdquo OpusculaMathematica vol 29 no 2 pp 177ndash186 2009

[5] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939

[6] T S Blyth and M F Janowitz Residuation Theory PergamonPress 1972

[7] T Kowalski and H Ono Residuated Lattices An AlgebraicGlimpse at Logic without Contraction Japan Advanced Instituteof Science and Technology March 2001

[8] D PiciuAlgebras of Fuzzy Logic Editura Universitaria CraiovaCraiova Romania 2007

[9] E TurunenMathematics Behind Fuzzy Logic Physica 1999[10] P Hajek Metamathematics of Fuzzy Logic vol 4 of Trends

in LogicmdashStudia Logica Library Kluwer Academic DordrechtThe Netherlands 1998

[11] Sh Ghorbani ldquoSub positive implicative filters of residuatedlatticesrdquo World Applied Science Journal vol 12 no 5 pp 586ndash590 2011

[12] S Ghorbani and A Hasankhani ldquoFuzzy convex subalgebrasof commutative residuated latticesrdquo Iranian Journal of FuzzySystems vol 7 no 2 pp 41ndash54 159 2010

[13] R Biswas ldquoVague groupsrdquo International Journal of Computa-tional Cognition vol 4 pp 20ndash23 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Discrete Mathematics 5

4 Lattice of Vague Filters

Let 119860 and 119861 be vague sets of a residuated lattice 119871 If 119881119860(119909) le

119881119861(119909) for all 119909 isin 119871 then we say 119861 containing119860 and denote it

by 119860 ⪯ 119861The unite vague set of a residuated lattice 119871 is a vague

filter of 119871 containing any vague filter of 119871 For a vague setwe can define the intersection of two vague sets 119860 and 119861 of aresiduated lattice 119871 by 119881

119860cap119861(119909) = imin119881

119860(119909) 119881119861(119909) for all

119909 isin 119871 It is easy to prove that the intersection of any family ofvague filters of 119871 is a vague filter of 119871 Hence we can define avague filter generated by a vague set as follows

Definition 25 Let 119860 be a vague set of a residuated lattice 119871A vague filter 119861 is called a vague filter generated by 119860 if itsatisfies

(i) 119860 ⪯ 119861(ii) 119860 ⪯ 119862 that implies 119861 ⪯ 119862 for all vague filters 119862 of 119871

The vague filter generated by 119860 is denoted by ⟨119860⟩

Example 26 Consider the vague lattice filter119860 in Example 20which is not a vague filter of 119871Then the vague filter generatedby 119860 is

⟨119860⟩ = ⟨0 [03 04]⟩ ⟨119886 [03 04]⟩

⟨119887 [03 04]⟩ ⟨119888 [03 08]⟩ ⟨0 [05 09]⟩

(14)

Theorem 27 Let 119860 be a vague set of a residuated lattice 119871Then the vague value of 119909 in ⟨119860⟩ is

119881⟨119860⟩ (119909) = isup imin VA (a1) VA (a2) VA (an)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

(15)

for all 119909 isin 119871

Proof First we will prove that ⟨119860⟩ is a vague set of 119871

119905⟨119860⟩ (119909)

= sup min 119905119860(1198861) 119905119860(1198862) 119905

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

le sup min 1 minus 119891119860(1198861) 1 minus 119891

119860(1198862) 1 minus 119891

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

= sup 1 minus max 119891119860(1198861) 119891119860(1198862) 119891

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

= 1 minus inf max 119891119860(1198861) 119891119860(1198862) 119891

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

= 1 minus 119891⟨119860⟩ (119909)

(16)

Now we will show that ⟨119860⟩ is a vague filter of 119871(VF1) Suppose that 119909 119910 isin 119871 is arbitrary For every 120576 gt 0

we can take 1198861 119886

119899 1198871 119887

119898isin 119871 such that119909 ge 119886

1lowastsdot sdot sdotlowast119886

119899

119910 ge 1198871lowast sdot sdot sdot lowast 119887

119898 and

119905⟨119860⟩ (119909) minus 120576 lt min 119905

119860(1198861) 119905119860(1198862) 119905

119860(119886119899)

119905⟨119860⟩

(119910) minus 120576 lt min 119905119860(1198871) 119905119860(1198872) 119905

119860(119887119898)

(17)

By Proposition 2 part (4) we have 119909 lowast 119910 ge 1198861lowast sdot sdot sdot lowast 119886

119899lowast 1198871lowast

sdot sdot sdot lowast 119887119898and then

119905⟨119860⟩

(119909 lowast 119910)

ge min 119905119860(1198861) 119905

119860(119886119899) 119905119860(1198871) 119905

119860(119887119898)

= min min 119905119860(1198861) 119905

119860(119886119899)

min 119905119860(1198871) 119905

119860(119887119898)

gt min (119905⟨119860⟩ (119909) minus 120576) (119905

⟨119860⟩(119910) minus 120576)

= min 119905⟨119860⟩ (119909) 119905⟨119860⟩ (119910) minus 120576

(18)

Since 120576 gt 0 is arbitrary we obtain that 119905⟨119860⟩

(119909 lowast 119910) ge

min119905⟨119860⟩

(119909) 119905⟨119860⟩

(119910)Similarly we can show that 1 minus 119891

⟨119860⟩(119909 lowast 119910) ge min1 minus

119891⟨119860⟩

(119909) 1 minus119891⟨119860⟩

(119910) Hence119881119860(119909lowast119910) ge imin119881

119860(119909) 119881119860(119910)

for all 119909 119910 isin 119871(VF2) Suppose that 119909 le 119910where 119909 119910 isin 119871 Let 119886

1 119886

119899isin

119871 such that 1198861lowast sdot sdot sdot lowast 119886

119899le 119909 Then 119886

1lowast sdot sdot sdot lowast 119886

119899le 119910 And we

get that

119881119860(119910) ge imin 119881

119860(1198861) 119881119860(1198862) 119881

119860(119886119899) (19)

Since 1198861 119886

119899isin 119871 where 119909 ge 119886

1lowast sdot sdot sdot lowast 119886

119899is arbitrary then

119881⟨119860⟩

(119910) ge isup imin 119881119860(1198861) 119881119860(1198862) 119881

119860(119886119899)

119909 ge 1198861lowast sdot sdot sdot lowast 119886

119899

= 119881⟨119860⟩ (119909)

(20)

Therefore ⟨119860⟩ is a vague filter of 119871 by Theorem 16 It is clearthat119860 ⪯ ⟨119860⟩ Now let119862 be a vague filter of119871 such that119860 ⪯ 119862Then we have

119881⟨119860⟩ (119909)

= isup imin 119881119860(1198861) 119881

119860(119886119899) 119909 ge 119886

1lowast sdot sdot sdot lowast 119886

119899

le isup imin 119881119862(1198861) 119881

119862(119886119899) 119909 ge 119886

1lowast sdot sdot sdot lowast 119886

119899

= 119881119862 (119909)

(21)

for all 119909 isin 119871 Hence ⟨119860⟩ ⪯ 119862 and then ⟨119860⟩ is a vague filtergenerated by 119860

We denote the set of all vague filters of a residuated lattice119871 by VF(119871)

6 Journal of Discrete Mathematics

Corollary 28 (VF(119871) cap or) is a complete lattice

Proof Clearly if 119860119894119894isinΓ

is a family of vague filters of aresiduated lattice 119871 then the infimumof this family is⋂

119894isinΓ119860119894

and the supremum is ⋁119894isinΓ

119860119894= ⟨⋃119894isinΓ

119860119894⟩ It is easy to prove

that (IFF(119871) cap or) is a complete lattice

Let 119860 and 119861 be vague sets of a residuated lattice 119871 Then119860 lowast 119861 is defined as follows

119881(119860lowast119861) (119911) = isup imin 119881

119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910 (22)

for all 119911 isin 119871

Theorem 29 Let 119860 and 119861 be vague sets of a residuated lattice119871 Then 119860 lowast 119861 is a vague set of 119871

Proof Since 119860 and 119861 are vague sets of 119871 we have 119905119860(119909) le

1 minus 119891119860(119909) and 119905

119861(119910) le 1 minus 119891

119861(119910) for all 119909 119910 isin 119871 Thus

119905119860lowast119861 (119911) = sup min 119905

119860 (119909) 119905119860 (119910) 119911 ge 119909 lowast 119910

le sup min 1 minus 119891119860 (119909) 1 minus 119891

119860(119910) 119911 ge 119909 lowast 119910

le 1 minus 119891119860lowast119861 (119911)

(23)

Hence 119905119860lowast119861

(119911) + 119891119860lowast119861

(119911) le 1 and then 119860 lowast 119861 is a vague set of119871

Theorem30 Let119860 and119861 be vague filters of a residuated lattice119871 Then 119860 lowast 119861 is a vague filter of 119871

Proof ByTheorem 29119860lowast119861 is a vague set of 119871 We will provethat 119860 lowast 119861 is a vague filter of 119871

(VF2) Let 1199111

le 1199112and 119909 lowast 119910 le 119911

1 We get that

imin119881119860(119909) 119881119861(119910) le 119881

119860lowast119861(1199112) Since 119909 lowast 119910 le 119911

1is arbitrary

we have

119881119860lowast119861

(1199111) = isup imin 119881

119860 (119909) 119881119861 (119910) 1199111ge 119909 lowast 119910

le 119881119860lowast119861

(1199112)

(24)

(VF1) Let 119909 119910 isin 119871 Then

imin 119881119860lowast119861 (119909) 119881119860lowast119861 (119910)

= imin isup imin 119881119860 (119886) 119881119861 (119887) 119909 ge 119886 lowast 119887

isup imin 119881119860 (119888) 119881119861 (119889) 119910 ge 119888 lowast 119889

= isup isup imin 119881119860 (119886 lowast 119888) 119881119861 (119888 lowast 119889)

119909 ge 119886 lowast 119887 119910 ge 119888 lowast 119889

le isup imin 119881119860 (119886 lowast 119888) 119881119861 (119888 lowast 119889)

119909 lowast 119910 ge (119886 lowast 119887) lowast (119888 lowast 119889)

le isup imin 119881119860 (119890) 119881119861 (119891) 119909 lowast 119910 ge 119890 lowast 119891

= 119881119860lowast119861

(119909 lowast 119910)

(25)

By Definition 13 119860 lowast 119861 is a vague filter of 119871

Theorem31 Let119860 and119861 be vague filters of a residuated lattice119871 such that119881

119860(1) = 119881

119861(1) Then119860lowast119861 is the least upper bound

of 119860 and 119861

Proof By Definition 13 we have 119881119860(1) = 119881

119861(1) ge 119881

119860(119910) for

all 119910 isin 119871 Let 119911 isin 119871 Since 119911 = 1 lowast 119911 we get that

119881119860lowast119861 (119911) = isup imin 119881

119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910

ge imin 119881119860 (119911) 119881119861 (1) ge 119881

119860 (119911)

(26)

Hence 119860 ⪯ 119860 lowast 119861 Similarly we can show that 119861 ⪯ 119860 lowast 119861Suppose that 119862 is the vague filter of 119871 containing 119860 and 119861Then

119881119860lowast119861 (119911)

= isup imin 119881119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910

le isup imin 119881119862 (119909) 119881119862 (119910) 119911 ge 119909 lowast 119910

le isup imin 119881119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 = 120583

119862 (119911)

(27)

Hence 119860lowast 119861 ⪯ 119862 that is 119860lowast 119861 is the least upper bound of 119860and 119861

Lemma 32 Let119862 be a vague filter of a residuated lattice 119871 and119911 isin 119871 Then

(1) 119881119862(119911) = isupVC(x lowast y) z ge x lowast y for all 119909 119910 119911 isin 119871

(2) 119881119862(119909 lowast 119910) = iminVC(x)VC(y) for 119909 119910 isin 119871

Proof (1) Since 119909 lowast 119910 le 119911 then 119881119862(119909 lowast 119910) le 119881

119862(119911) by (VF2)

Hence 119881119862(119911) ge isup119881

119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 Also we have

1 lowast 119911 = 119911 We obtain that

119881119862 (119911) = 119881

119862 (1 lowast 119911) le isup 119881119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 (28)

(2) It follows from Proposition 2 part (2) andDefinition 13

Let VFD(119871) be the set of all vague filters119860 and119861 of 119871 suchthat 119881

119860(1) = 119881

119861(1)

Theorem 33 (119881119865119863(119871) cap lowast) is a distributive lattice

Journal of Discrete Mathematics 7

Proof Let 119860 119861 119862 isin VFD(119871) be such that 119860 ⪯ 119862 We willshow that (119860 lowast 119861) cap 119862 = (119860 lowast 119862) cap (119861 lowast 119862) By Lemma 32

119881(119860lowast119861)cap119862 (119911)

= imin (119881119860lowast119861

) (119911) 119881119862 (119911)

= imin isup imin 119881119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910 119881

119862 (119911)

= isup imin 119881119860 (119909) 119881119861 (119910) 119881

119862 (119911) 119911 ge 119909 lowast 119910

= isup imin 119881119860 (119909) 119881119861 (119910) 119881

119862(119909 lowast 119910) 119911 ge 119909 lowast 119910

= isup imin 119881119860 (119909) 119881119861 (119910) 119881

119862 (119909) 119881119862 (119910) 119911 ge 119909 lowast 119910

= isup imin 119881119860 (119909) 119881119862 (119909)

imin 119881119861(119910) 119881

119862(119910) 119911 ge 119909 lowast 119910

= isup imin 119881119860cap119862 (119909) 119881119861cap119862 (119910) 119911 ge 119909 lowast 119910

= 119881(119860cap119862)lowast(119861cap119862) (119911)

(29)

5 Subpositive Implicative Vague Filters

Definition 34 Let 119860 be a vague subset of a residuated lattice119871 119860 is called a subpositive implicative vague filter of 119871 if

(VF6) 119881119860(119909) le 119881

119860(1) for all 119909 isin 119871

(SVF1) imin119881119860(((119909 rarr 119910) lowast 119911) 999492999484 ((119910 999492999484 119909) rarr

119909)) 119881119860(119911) le 119881

119860((119909 rarr 119910) 999492999484 119910) for any

119909 119910 119911 isin 119871(SVF2) imin119881

119860((119911 lowast (119909 999492999484 119910)) rarr ((119910 rarr 119909) 999492999484

119909)) 119881119860(119911) le 119881

119860((119909 999492999484 119910) rarr 119910) for any

119909 119910 119911 isin 119871

Proposition 35 Let 119860 be a vague filter of a residuated lattice119871 Then 119860 is a subpositive implicative vague filter of L if andonly if it satisfies

(SVF3) 119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) le

119881119860((119909 rarr 119910) 999492999484 119910) for any 119909 119910 isin 119871

(SVF4) 119881119860((119909 999492999484 119910) rarr ((119910 rarr 119909) 999492999484 119909)) le

119881119860((119909 999492999484 119910) rarr 119910) for any 119909 119910 isin 119871

Proposition 36 Let119860 be a subpositive implicative vague filterof a residuated lattice 119871 Then 119860 is a vague filter of 119871

Proof We have imin119881119860(119909 999492999484 119910) 119881

119860(119909) = imin119881

119860((119910 rarr

119910) lowast 119909) 999492999484 ((119910 999492999484 119910) rarr 119910) 119881119860(119909) le 119881

119860((119910 rarr 119910) 999492999484

119910) = 119881119860(119910) By (VS1) andTheorem 17 119860 is a vague filter of a

residuated lattice 119871

The relation between subpositive implicative vague filterand its level subset is as follows

Theorem 37 Let 119860 be a vague set of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and only if

for all120572 120573 isin [0 1] the set119860(120572120573)

is either empty or a subpositiveimplicative filter of 119871 where 120572 le 120573

Proof Suppose that119860 is a subpositive implicative filter vagueset of 119871 and 120572 120573 isin [0 1] Let 119860

(120572120573)= 0 Since 119860 is a vague

filter then119860(120572120573)

is a filter of 119871 byTheorem 21 Now let (119909 rarr

119910) 999492999484 ((119910 999492999484 119909) rarr 119909) isin 119860(120572120573)

Then 119881119860((119909 rarr 119910) 999492999484

((119910 999492999484 119909) rarr 119909)) ge [120572 120573] By (SVF3) 119881119860((119909 rarr 119910) 999492999484

119910) ge [120572 120573] that is (119909 rarr 119910) 999492999484 119910 isin 119860(120572120573)

Similarly we canshow that119860

(120572120573)satisfies (SVF4) Hence119860

(120572120573)is a subpositive

implicative filter of 119871Conversely let 119905

119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) =

120572 and 1 minus 119891119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) = 120573 Thus

119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) ge [120572 120573] We get that

(119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909) isin 119860(120572120573)

So 119860(120572120573)

= 0Thus (119909 rarr 119910) 999492999484 119910 isin 119860

(120572120573) that is 119881

119860((119909 rarr 119910) 999492999484 119910) ge

[120572 120573] = 119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) Similarly we can

prove that 119860 satisfies (SVF4)

In the following theorems we obtain some characteriza-tions of subpositive implicative vague filter

Theorem 38 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF5) iminVA((y rarr z) 999492999484 (x rarr y))VA(x) le

VA(y) for any 119909 119910 119911 isin 119871(SVF6) iminVA((y 999492999484 z) rarr (x 999492999484 y))VA(x) le

VA(y) for any 119909 119910 119911 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871 ByProposition 2 part (3) we have imin119881

119860((119910 rarr 119911) 999492999484 (119909 rarr

119910)) 119881119860(119909) = imin119881

119860(119909 rarr ((119910 rarr 119911) 999492999484 119910)) 119881

119860(119909) le

119881119860((119910 rarr 119911) 999492999484 119910) By Proposition 2 part (7) (119910 rarr 119911) 999492999484

119910 le 119911 999492999484 119910 le (119910 rarr 119911) 999492999484 ((119911 999492999484 119910) rarr 119910) We get that119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119911 999492999484 119910) le 119881

119860((119910 rarr 119911) 999492999484 ((119911 999492999484

119910) rarr 119910)) le 119881119860((119911 999492999484 119910) rarr 119910) Hence imin119881

119860((119910 rarr

119911) 999492999484 119910) 119881119860(119911 999492999484 119910) le imin119881

119860((119911 999492999484 119910) rarr 119910) 119881

119860(119911 999492999484

119910) le 119881119860(119910) Thus it satisfies (SVF5) Similarly we can

prove (SVF6) Conversely let 119860 satisfy (SVF5) and (SVF6)119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) = imin119881

119860([((119909 rarr 119910) 999492999484

119910) rarr 1] 999492999484 [(119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909) rarr ((119909 rarr

119910) 999492999484 119910)]) 119881119860(1) le 119881

119860((119909 rarr 119910) 999492999484 119910) Similarly we

can prove that (SVF5) holds Hence 119860 is a vague subpositiveimplicative filter

Theorem 39 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF7) 119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) le

119881119860((119910 999492999484 119909) rarr 119909) for any 119909 119910 isin 119871

(SVF8) 119881119860((119909 999492999484 119910) rarr ((119910 999492999484 119909) rarr 119909)) le

119881119860((119910 rarr 119909) 999492999484 119909) for any 119909 119910 isin 119871

Proof Let (SVF7) and (SVF8) hold We will show that 119860

satisfies (SVF5) and (SVF6) We have imin119881119860((119910 rarr 119911) 999492999484

(119909 rarr 119910)) 119881119860(119909) = imin119881

119860(119909 rarr ((119910 rarr 119911) 999492999484

119910)) 119881119860(119909) le 119881

119860((119910 rarr 119911) 999492999484 119910) le 119881

119860((119910 rarr 119911) 999492999484

8 Journal of Discrete Mathematics

((119911 rarr 119910) 999492999484 119910)) le 119881119860((119911 999492999484 119910) rarr 119910)) Also we have

119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119911 999492999484 119910) Since 119860 is a vague

filter 119881119860((119910 rarr 119911) 999492999484 119910) le imin119881

119860(119911 999492999484 119910) 119881

119860((119911 999492999484

119910) rarr 119910) le 119881119860(119910) Hence 119860 satisfies (SVF5) Analogously

we can prove that119860 satisfies (SVF6) Hence119860 is a subpositiveimplicative vague filter of 119871 byTheorem 38 Conversely let 119860be a subpositive implicative vague filter of 119871 We have

[((119910 999492999484 119909) 997888rarr 119909) 997888rarr 1]

999492999484 [((119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909))

997888rarr ((119910 999492999484 119909) 997888rarr 119909) ]

= [(119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)]

997888rarr [((119910 997888rarr 119909) 999492999484 119909) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)]

= (119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)

(30)

We get that 119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) =

imin119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909))119881

119860([((119910 999492999484 119909) rarr

119909) rarr 1] 999492999484 [((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) rarr ((119910 999492999484

119909) rarr 119909)]) le 119881119860((119910 999492999484 119909) rarr 119909) by (SVF5) Similarly we

can prove that (SVF8) holds

Theorem 40 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF9) 119881119860((119909 rarr 119910) 999492999484 119909) le 119881

119860(119909) for any 119909 119910 isin 119871

(SVF10) 119881119860((119909 999492999484 119910) rarr 119909) le 119881

119860(119909) for any 119909 119910 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871Then 119860 satisfies (SVF5) and (SVF6)

119881119860((119909 997888rarr 119910) 999492999484 119909)

= imin 119881119860(1 997888rarr ((119909 997888rarr 119910) 999492999484 119909)) 119881

119860 (1)

= imin 119881119860((119909 997888rarr 119910) 999492999484 (1 997888rarr 119909)) 119881119860 (1)

le 119881119860 (119909)

(31)

Similarly we can prove that (SVF10) holds Conversely let119860 satisfy (SVF9) and (SVF10) We will show that 119860 satisfies(SVF5) and (SVF6) imin119881

119860((119910 rarr 119911) 999492999484 (119909 rarr

119910)) 119881119860(119909) = imin119881

119860((119909 rarr (119910 rarr 119911) 999492999484 119910)) 119881

119860(119909) le

119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119910) Hence 119860 satisfies (SVF5)

Similarly we can prove (SVF7) holds Therefore 119860 is asubpositive implicative vague filter of L byTheorem 38

Theorem 41 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF11) 119881119860(not119909 999492999484 119909) le 119881

119860(119909) for any 119909 isin 119871

(SVF12) 119881119860(sim 119909 rarr 119909) le 119881

119860(119909) for any 119909 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871Then (SVF11) and (SVF12) follow by (SVF9) and (SVF10)Conversely since 0 le 119910 then 119909 rarr 0 le 119909 rarr 119910 Hence

(119909 rarr 119910) 999492999484 119909 le not119909 999492999484 119909 by Proposition 2 part (6)Therefore 119881

119860((119909 rarr 119910) 999492999484 119909) le 119881

119860(not119909 999492999484 119909) le 119881

119860(119909)

Similarly we can prove that (SVF10) holds Hence 119860 is asubpositive implicative vague filter of 119871 byTheorem 40

The extension theorem of subpositive implicative vaguefilters is obtained from the following theorem

Theorem 42 Let 119860 and 119861 be two vague filters of a residuatedlattice 119871 which satisfies 119860 le 119861 and 119881

119860(1) = 119881

119861(1) If 119860 is a

subpositive implicative vague filter of L so 119861 is

Proof Suppose that 119906 = (119909 rarr 119910) 999492999484 119909 Then

1 = 119906 997888rarr 119906 = 119906 997888rarr ((119909 997888rarr 119910) 999492999484 119909)

= (119909 997888rarr 119910) 999492999484 (119906 997888rarr 119909) = 1

(32)

We have (119909 rarr 119910) 999492999484 (119906 rarr 119909) le ((119906 rarr 119909) rarr 119910) 999492999484

(119906 rarr 119909) Therefore 119881119860(((119906 rarr 119909) rarr 119910) 999492999484 (119906 rarr 119909)) =

119881119860(1) By (SVF10) 119881

119860(1) = 119881

119860(((119906 rarr 119909) rarr 119910) 999492999484 (119906 rarr

119909)) le 119881119860(119906 rarr 119909) Hence 119881

119860(119906 rarr 119909) = 119881

119860(1) = 119881

119861(1)

Since 119860 le 119861 then 119881119861(119906 rarr 119909) ge 119881

119860(119906 rarr 119909) = 119881

119861(1)

We get that 119881119861((119909 rarr 119910) 999492999484 119909) = 119881

119861(119906) = imin119881

119861(119906 rarr

119909) 119881119861(119906) le 119881

119861(119909) Similarly we can show that 119866 satisfies

(SVF11) Hence 119866 is a vague subpositive implicative filter of119871 by Theorem 40

Definition 43 Let 119860 be a vague filter of a residuated lattice 119871119860 is called a vague Boolean filter of 119871 if

(BVF1) 119881119860(119909 or not119909) = 119881

119860(1) for all 119909 isin 119871

(BVF2) 119881119860(119909or sim 119909) = 119881

119860(1) for all 119909 isin 119871

Similar to Theorem 40 we have the following theoremwith respect to Boolean vague filters

Theorem 44 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a Boolean vague filter of 119871 if and only if for all120572 120573 isin [0 1] the set 119860

(120572120573)is either empty or a Boolean filter of

119871 where 120572 le 120573

Theorem 45 Let 119860 be a vague filter of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and if onlyif 119860 is a Boolean vague filter

Proof Let119860 be a subpositive implicative vague filterWe have

119881119860 (1) = 119881

119860 (not (119909 or not119909) 999492999484 (119909 or not119909))

= 119881119860 ((not119909 and notnot119909) 999492999484 (119909 or not119909)) le 119881

119860 (119909 or not119909)

(33)

Hence 119881119860(119909 or not119909) = 119881

119860(1) Similarly we can prove that 119860

satisfies (BVF2) Hence 119860 is a Boolean vague filterConversely let119881

119860(119909ornot119909) = 119881

119860(1)We have119881

119860(119909ornot119909) =

119881119860((119909 999492999484 119909) or (not119909 999492999484 119909))119881

119860((119909 or not119909) 999492999484 119909) = imin119881

119860((119909 or

not119909) 999492999484 119909) 119881119860(1) = imin119881

119860((119909 or not119909) 999492999484 119909) 119881

119860(119909 or not119909) le

119881119860(119909) Similarly we can show that119860 satisfies (SVF12) Hence

119860 is a subpositive implicative vague filter byTheorem 41

Journal of Discrete Mathematics 9

6 Conclusion

In this paper we introduced the notions of vague filterssubpositive implicative vague filters and Boolean vaguefilters of a residuated lattice and investigated some relatedproperties We studied the relationship between filters andvague filters of a residuated lattice and showed that vaguefilters of a residuated lattice is a generalization of fuzzy filtersWe proved that the set of all vague filters of a residuated latticeforms a complete lattice and obtained some characterizationsof subpositive implicative vague filterThe extension theoremof subpositive implicative vague filters was obtained Finallyit was proved that subpositive implicative vague filters areequivalent to Boolean vague filters

Conflict of Interests

There is no conflict of interests regarding the publication ofthis paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993

[3] Y B Jun and C H Park ldquoIdeals of pseudo MV-algebras basedon vague set theoryrdquo Iranian Journal of Fuzzy Systems vol 6 no7 pp 31ndash47 2009

[4] A Borumand Saeid ldquoVague BCKBCI-algebrasrdquo OpusculaMathematica vol 29 no 2 pp 177ndash186 2009

[5] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939

[6] T S Blyth and M F Janowitz Residuation Theory PergamonPress 1972

[7] T Kowalski and H Ono Residuated Lattices An AlgebraicGlimpse at Logic without Contraction Japan Advanced Instituteof Science and Technology March 2001

[8] D PiciuAlgebras of Fuzzy Logic Editura Universitaria CraiovaCraiova Romania 2007

[9] E TurunenMathematics Behind Fuzzy Logic Physica 1999[10] P Hajek Metamathematics of Fuzzy Logic vol 4 of Trends

in LogicmdashStudia Logica Library Kluwer Academic DordrechtThe Netherlands 1998

[11] Sh Ghorbani ldquoSub positive implicative filters of residuatedlatticesrdquo World Applied Science Journal vol 12 no 5 pp 586ndash590 2011

[12] S Ghorbani and A Hasankhani ldquoFuzzy convex subalgebrasof commutative residuated latticesrdquo Iranian Journal of FuzzySystems vol 7 no 2 pp 41ndash54 159 2010

[13] R Biswas ldquoVague groupsrdquo International Journal of Computa-tional Cognition vol 4 pp 20ndash23 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Discrete Mathematics

Corollary 28 (VF(119871) cap or) is a complete lattice

Proof Clearly if 119860119894119894isinΓ

is a family of vague filters of aresiduated lattice 119871 then the infimumof this family is⋂

119894isinΓ119860119894

and the supremum is ⋁119894isinΓ

119860119894= ⟨⋃119894isinΓ

119860119894⟩ It is easy to prove

that (IFF(119871) cap or) is a complete lattice

Let 119860 and 119861 be vague sets of a residuated lattice 119871 Then119860 lowast 119861 is defined as follows

119881(119860lowast119861) (119911) = isup imin 119881

119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910 (22)

for all 119911 isin 119871

Theorem 29 Let 119860 and 119861 be vague sets of a residuated lattice119871 Then 119860 lowast 119861 is a vague set of 119871

Proof Since 119860 and 119861 are vague sets of 119871 we have 119905119860(119909) le

1 minus 119891119860(119909) and 119905

119861(119910) le 1 minus 119891

119861(119910) for all 119909 119910 isin 119871 Thus

119905119860lowast119861 (119911) = sup min 119905

119860 (119909) 119905119860 (119910) 119911 ge 119909 lowast 119910

le sup min 1 minus 119891119860 (119909) 1 minus 119891

119860(119910) 119911 ge 119909 lowast 119910

le 1 minus 119891119860lowast119861 (119911)

(23)

Hence 119905119860lowast119861

(119911) + 119891119860lowast119861

(119911) le 1 and then 119860 lowast 119861 is a vague set of119871

Theorem30 Let119860 and119861 be vague filters of a residuated lattice119871 Then 119860 lowast 119861 is a vague filter of 119871

Proof ByTheorem 29119860lowast119861 is a vague set of 119871 We will provethat 119860 lowast 119861 is a vague filter of 119871

(VF2) Let 1199111

le 1199112and 119909 lowast 119910 le 119911

1 We get that

imin119881119860(119909) 119881119861(119910) le 119881

119860lowast119861(1199112) Since 119909 lowast 119910 le 119911

1is arbitrary

we have

119881119860lowast119861

(1199111) = isup imin 119881

119860 (119909) 119881119861 (119910) 1199111ge 119909 lowast 119910

le 119881119860lowast119861

(1199112)

(24)

(VF1) Let 119909 119910 isin 119871 Then

imin 119881119860lowast119861 (119909) 119881119860lowast119861 (119910)

= imin isup imin 119881119860 (119886) 119881119861 (119887) 119909 ge 119886 lowast 119887

isup imin 119881119860 (119888) 119881119861 (119889) 119910 ge 119888 lowast 119889

= isup isup imin 119881119860 (119886 lowast 119888) 119881119861 (119888 lowast 119889)

119909 ge 119886 lowast 119887 119910 ge 119888 lowast 119889

le isup imin 119881119860 (119886 lowast 119888) 119881119861 (119888 lowast 119889)

119909 lowast 119910 ge (119886 lowast 119887) lowast (119888 lowast 119889)

le isup imin 119881119860 (119890) 119881119861 (119891) 119909 lowast 119910 ge 119890 lowast 119891

= 119881119860lowast119861

(119909 lowast 119910)

(25)

By Definition 13 119860 lowast 119861 is a vague filter of 119871

Theorem31 Let119860 and119861 be vague filters of a residuated lattice119871 such that119881

119860(1) = 119881

119861(1) Then119860lowast119861 is the least upper bound

of 119860 and 119861

Proof By Definition 13 we have 119881119860(1) = 119881

119861(1) ge 119881

119860(119910) for

all 119910 isin 119871 Let 119911 isin 119871 Since 119911 = 1 lowast 119911 we get that

119881119860lowast119861 (119911) = isup imin 119881

119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910

ge imin 119881119860 (119911) 119881119861 (1) ge 119881

119860 (119911)

(26)

Hence 119860 ⪯ 119860 lowast 119861 Similarly we can show that 119861 ⪯ 119860 lowast 119861Suppose that 119862 is the vague filter of 119871 containing 119860 and 119861Then

119881119860lowast119861 (119911)

= isup imin 119881119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910

le isup imin 119881119862 (119909) 119881119862 (119910) 119911 ge 119909 lowast 119910

le isup imin 119881119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 = 120583

119862 (119911)

(27)

Hence 119860lowast 119861 ⪯ 119862 that is 119860lowast 119861 is the least upper bound of 119860and 119861

Lemma 32 Let119862 be a vague filter of a residuated lattice 119871 and119911 isin 119871 Then

(1) 119881119862(119911) = isupVC(x lowast y) z ge x lowast y for all 119909 119910 119911 isin 119871

(2) 119881119862(119909 lowast 119910) = iminVC(x)VC(y) for 119909 119910 isin 119871

Proof (1) Since 119909 lowast 119910 le 119911 then 119881119862(119909 lowast 119910) le 119881

119862(119911) by (VF2)

Hence 119881119862(119911) ge isup119881

119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 Also we have

1 lowast 119911 = 119911 We obtain that

119881119862 (119911) = 119881

119862 (1 lowast 119911) le isup 119881119862(119909 lowast 119910) 119911 ge 119909 lowast 119910 (28)

(2) It follows from Proposition 2 part (2) andDefinition 13

Let VFD(119871) be the set of all vague filters119860 and119861 of 119871 suchthat 119881

119860(1) = 119881

119861(1)

Theorem 33 (119881119865119863(119871) cap lowast) is a distributive lattice

Journal of Discrete Mathematics 7

Proof Let 119860 119861 119862 isin VFD(119871) be such that 119860 ⪯ 119862 We willshow that (119860 lowast 119861) cap 119862 = (119860 lowast 119862) cap (119861 lowast 119862) By Lemma 32

119881(119860lowast119861)cap119862 (119911)

= imin (119881119860lowast119861

) (119911) 119881119862 (119911)

= imin isup imin 119881119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910 119881

119862 (119911)

= isup imin 119881119860 (119909) 119881119861 (119910) 119881

119862 (119911) 119911 ge 119909 lowast 119910

= isup imin 119881119860 (119909) 119881119861 (119910) 119881

119862(119909 lowast 119910) 119911 ge 119909 lowast 119910

= isup imin 119881119860 (119909) 119881119861 (119910) 119881

119862 (119909) 119881119862 (119910) 119911 ge 119909 lowast 119910

= isup imin 119881119860 (119909) 119881119862 (119909)

imin 119881119861(119910) 119881

119862(119910) 119911 ge 119909 lowast 119910

= isup imin 119881119860cap119862 (119909) 119881119861cap119862 (119910) 119911 ge 119909 lowast 119910

= 119881(119860cap119862)lowast(119861cap119862) (119911)

(29)

5 Subpositive Implicative Vague Filters

Definition 34 Let 119860 be a vague subset of a residuated lattice119871 119860 is called a subpositive implicative vague filter of 119871 if

(VF6) 119881119860(119909) le 119881

119860(1) for all 119909 isin 119871

(SVF1) imin119881119860(((119909 rarr 119910) lowast 119911) 999492999484 ((119910 999492999484 119909) rarr

119909)) 119881119860(119911) le 119881

119860((119909 rarr 119910) 999492999484 119910) for any

119909 119910 119911 isin 119871(SVF2) imin119881

119860((119911 lowast (119909 999492999484 119910)) rarr ((119910 rarr 119909) 999492999484

119909)) 119881119860(119911) le 119881

119860((119909 999492999484 119910) rarr 119910) for any

119909 119910 119911 isin 119871

Proposition 35 Let 119860 be a vague filter of a residuated lattice119871 Then 119860 is a subpositive implicative vague filter of L if andonly if it satisfies

(SVF3) 119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) le

119881119860((119909 rarr 119910) 999492999484 119910) for any 119909 119910 isin 119871

(SVF4) 119881119860((119909 999492999484 119910) rarr ((119910 rarr 119909) 999492999484 119909)) le

119881119860((119909 999492999484 119910) rarr 119910) for any 119909 119910 isin 119871

Proposition 36 Let119860 be a subpositive implicative vague filterof a residuated lattice 119871 Then 119860 is a vague filter of 119871

Proof We have imin119881119860(119909 999492999484 119910) 119881

119860(119909) = imin119881

119860((119910 rarr

119910) lowast 119909) 999492999484 ((119910 999492999484 119910) rarr 119910) 119881119860(119909) le 119881

119860((119910 rarr 119910) 999492999484

119910) = 119881119860(119910) By (VS1) andTheorem 17 119860 is a vague filter of a

residuated lattice 119871

The relation between subpositive implicative vague filterand its level subset is as follows

Theorem 37 Let 119860 be a vague set of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and only if

for all120572 120573 isin [0 1] the set119860(120572120573)

is either empty or a subpositiveimplicative filter of 119871 where 120572 le 120573

Proof Suppose that119860 is a subpositive implicative filter vagueset of 119871 and 120572 120573 isin [0 1] Let 119860

(120572120573)= 0 Since 119860 is a vague

filter then119860(120572120573)

is a filter of 119871 byTheorem 21 Now let (119909 rarr

119910) 999492999484 ((119910 999492999484 119909) rarr 119909) isin 119860(120572120573)

Then 119881119860((119909 rarr 119910) 999492999484

((119910 999492999484 119909) rarr 119909)) ge [120572 120573] By (SVF3) 119881119860((119909 rarr 119910) 999492999484

119910) ge [120572 120573] that is (119909 rarr 119910) 999492999484 119910 isin 119860(120572120573)

Similarly we canshow that119860

(120572120573)satisfies (SVF4) Hence119860

(120572120573)is a subpositive

implicative filter of 119871Conversely let 119905

119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) =

120572 and 1 minus 119891119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) = 120573 Thus

119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) ge [120572 120573] We get that

(119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909) isin 119860(120572120573)

So 119860(120572120573)

= 0Thus (119909 rarr 119910) 999492999484 119910 isin 119860

(120572120573) that is 119881

119860((119909 rarr 119910) 999492999484 119910) ge

[120572 120573] = 119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) Similarly we can

prove that 119860 satisfies (SVF4)

In the following theorems we obtain some characteriza-tions of subpositive implicative vague filter

Theorem 38 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF5) iminVA((y rarr z) 999492999484 (x rarr y))VA(x) le

VA(y) for any 119909 119910 119911 isin 119871(SVF6) iminVA((y 999492999484 z) rarr (x 999492999484 y))VA(x) le

VA(y) for any 119909 119910 119911 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871 ByProposition 2 part (3) we have imin119881

119860((119910 rarr 119911) 999492999484 (119909 rarr

119910)) 119881119860(119909) = imin119881

119860(119909 rarr ((119910 rarr 119911) 999492999484 119910)) 119881

119860(119909) le

119881119860((119910 rarr 119911) 999492999484 119910) By Proposition 2 part (7) (119910 rarr 119911) 999492999484

119910 le 119911 999492999484 119910 le (119910 rarr 119911) 999492999484 ((119911 999492999484 119910) rarr 119910) We get that119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119911 999492999484 119910) le 119881

119860((119910 rarr 119911) 999492999484 ((119911 999492999484

119910) rarr 119910)) le 119881119860((119911 999492999484 119910) rarr 119910) Hence imin119881

119860((119910 rarr

119911) 999492999484 119910) 119881119860(119911 999492999484 119910) le imin119881

119860((119911 999492999484 119910) rarr 119910) 119881

119860(119911 999492999484

119910) le 119881119860(119910) Thus it satisfies (SVF5) Similarly we can

prove (SVF6) Conversely let 119860 satisfy (SVF5) and (SVF6)119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) = imin119881

119860([((119909 rarr 119910) 999492999484

119910) rarr 1] 999492999484 [(119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909) rarr ((119909 rarr

119910) 999492999484 119910)]) 119881119860(1) le 119881

119860((119909 rarr 119910) 999492999484 119910) Similarly we

can prove that (SVF5) holds Hence 119860 is a vague subpositiveimplicative filter

Theorem 39 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF7) 119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) le

119881119860((119910 999492999484 119909) rarr 119909) for any 119909 119910 isin 119871

(SVF8) 119881119860((119909 999492999484 119910) rarr ((119910 999492999484 119909) rarr 119909)) le

119881119860((119910 rarr 119909) 999492999484 119909) for any 119909 119910 isin 119871

Proof Let (SVF7) and (SVF8) hold We will show that 119860

satisfies (SVF5) and (SVF6) We have imin119881119860((119910 rarr 119911) 999492999484

(119909 rarr 119910)) 119881119860(119909) = imin119881

119860(119909 rarr ((119910 rarr 119911) 999492999484

119910)) 119881119860(119909) le 119881

119860((119910 rarr 119911) 999492999484 119910) le 119881

119860((119910 rarr 119911) 999492999484

8 Journal of Discrete Mathematics

((119911 rarr 119910) 999492999484 119910)) le 119881119860((119911 999492999484 119910) rarr 119910)) Also we have

119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119911 999492999484 119910) Since 119860 is a vague

filter 119881119860((119910 rarr 119911) 999492999484 119910) le imin119881

119860(119911 999492999484 119910) 119881

119860((119911 999492999484

119910) rarr 119910) le 119881119860(119910) Hence 119860 satisfies (SVF5) Analogously

we can prove that119860 satisfies (SVF6) Hence119860 is a subpositiveimplicative vague filter of 119871 byTheorem 38 Conversely let 119860be a subpositive implicative vague filter of 119871 We have

[((119910 999492999484 119909) 997888rarr 119909) 997888rarr 1]

999492999484 [((119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909))

997888rarr ((119910 999492999484 119909) 997888rarr 119909) ]

= [(119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)]

997888rarr [((119910 997888rarr 119909) 999492999484 119909) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)]

= (119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)

(30)

We get that 119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) =

imin119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909))119881

119860([((119910 999492999484 119909) rarr

119909) rarr 1] 999492999484 [((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) rarr ((119910 999492999484

119909) rarr 119909)]) le 119881119860((119910 999492999484 119909) rarr 119909) by (SVF5) Similarly we

can prove that (SVF8) holds

Theorem 40 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF9) 119881119860((119909 rarr 119910) 999492999484 119909) le 119881

119860(119909) for any 119909 119910 isin 119871

(SVF10) 119881119860((119909 999492999484 119910) rarr 119909) le 119881

119860(119909) for any 119909 119910 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871Then 119860 satisfies (SVF5) and (SVF6)

119881119860((119909 997888rarr 119910) 999492999484 119909)

= imin 119881119860(1 997888rarr ((119909 997888rarr 119910) 999492999484 119909)) 119881

119860 (1)

= imin 119881119860((119909 997888rarr 119910) 999492999484 (1 997888rarr 119909)) 119881119860 (1)

le 119881119860 (119909)

(31)

Similarly we can prove that (SVF10) holds Conversely let119860 satisfy (SVF9) and (SVF10) We will show that 119860 satisfies(SVF5) and (SVF6) imin119881

119860((119910 rarr 119911) 999492999484 (119909 rarr

119910)) 119881119860(119909) = imin119881

119860((119909 rarr (119910 rarr 119911) 999492999484 119910)) 119881

119860(119909) le

119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119910) Hence 119860 satisfies (SVF5)

Similarly we can prove (SVF7) holds Therefore 119860 is asubpositive implicative vague filter of L byTheorem 38

Theorem 41 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF11) 119881119860(not119909 999492999484 119909) le 119881

119860(119909) for any 119909 isin 119871

(SVF12) 119881119860(sim 119909 rarr 119909) le 119881

119860(119909) for any 119909 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871Then (SVF11) and (SVF12) follow by (SVF9) and (SVF10)Conversely since 0 le 119910 then 119909 rarr 0 le 119909 rarr 119910 Hence

(119909 rarr 119910) 999492999484 119909 le not119909 999492999484 119909 by Proposition 2 part (6)Therefore 119881

119860((119909 rarr 119910) 999492999484 119909) le 119881

119860(not119909 999492999484 119909) le 119881

119860(119909)

Similarly we can prove that (SVF10) holds Hence 119860 is asubpositive implicative vague filter of 119871 byTheorem 40

The extension theorem of subpositive implicative vaguefilters is obtained from the following theorem

Theorem 42 Let 119860 and 119861 be two vague filters of a residuatedlattice 119871 which satisfies 119860 le 119861 and 119881

119860(1) = 119881

119861(1) If 119860 is a

subpositive implicative vague filter of L so 119861 is

Proof Suppose that 119906 = (119909 rarr 119910) 999492999484 119909 Then

1 = 119906 997888rarr 119906 = 119906 997888rarr ((119909 997888rarr 119910) 999492999484 119909)

= (119909 997888rarr 119910) 999492999484 (119906 997888rarr 119909) = 1

(32)

We have (119909 rarr 119910) 999492999484 (119906 rarr 119909) le ((119906 rarr 119909) rarr 119910) 999492999484

(119906 rarr 119909) Therefore 119881119860(((119906 rarr 119909) rarr 119910) 999492999484 (119906 rarr 119909)) =

119881119860(1) By (SVF10) 119881

119860(1) = 119881

119860(((119906 rarr 119909) rarr 119910) 999492999484 (119906 rarr

119909)) le 119881119860(119906 rarr 119909) Hence 119881

119860(119906 rarr 119909) = 119881

119860(1) = 119881

119861(1)

Since 119860 le 119861 then 119881119861(119906 rarr 119909) ge 119881

119860(119906 rarr 119909) = 119881

119861(1)

We get that 119881119861((119909 rarr 119910) 999492999484 119909) = 119881

119861(119906) = imin119881

119861(119906 rarr

119909) 119881119861(119906) le 119881

119861(119909) Similarly we can show that 119866 satisfies

(SVF11) Hence 119866 is a vague subpositive implicative filter of119871 by Theorem 40

Definition 43 Let 119860 be a vague filter of a residuated lattice 119871119860 is called a vague Boolean filter of 119871 if

(BVF1) 119881119860(119909 or not119909) = 119881

119860(1) for all 119909 isin 119871

(BVF2) 119881119860(119909or sim 119909) = 119881

119860(1) for all 119909 isin 119871

Similar to Theorem 40 we have the following theoremwith respect to Boolean vague filters

Theorem 44 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a Boolean vague filter of 119871 if and only if for all120572 120573 isin [0 1] the set 119860

(120572120573)is either empty or a Boolean filter of

119871 where 120572 le 120573

Theorem 45 Let 119860 be a vague filter of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and if onlyif 119860 is a Boolean vague filter

Proof Let119860 be a subpositive implicative vague filterWe have

119881119860 (1) = 119881

119860 (not (119909 or not119909) 999492999484 (119909 or not119909))

= 119881119860 ((not119909 and notnot119909) 999492999484 (119909 or not119909)) le 119881

119860 (119909 or not119909)

(33)

Hence 119881119860(119909 or not119909) = 119881

119860(1) Similarly we can prove that 119860

satisfies (BVF2) Hence 119860 is a Boolean vague filterConversely let119881

119860(119909ornot119909) = 119881

119860(1)We have119881

119860(119909ornot119909) =

119881119860((119909 999492999484 119909) or (not119909 999492999484 119909))119881

119860((119909 or not119909) 999492999484 119909) = imin119881

119860((119909 or

not119909) 999492999484 119909) 119881119860(1) = imin119881

119860((119909 or not119909) 999492999484 119909) 119881

119860(119909 or not119909) le

119881119860(119909) Similarly we can show that119860 satisfies (SVF12) Hence

119860 is a subpositive implicative vague filter byTheorem 41

Journal of Discrete Mathematics 9

6 Conclusion

In this paper we introduced the notions of vague filterssubpositive implicative vague filters and Boolean vaguefilters of a residuated lattice and investigated some relatedproperties We studied the relationship between filters andvague filters of a residuated lattice and showed that vaguefilters of a residuated lattice is a generalization of fuzzy filtersWe proved that the set of all vague filters of a residuated latticeforms a complete lattice and obtained some characterizationsof subpositive implicative vague filterThe extension theoremof subpositive implicative vague filters was obtained Finallyit was proved that subpositive implicative vague filters areequivalent to Boolean vague filters

Conflict of Interests

There is no conflict of interests regarding the publication ofthis paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993

[3] Y B Jun and C H Park ldquoIdeals of pseudo MV-algebras basedon vague set theoryrdquo Iranian Journal of Fuzzy Systems vol 6 no7 pp 31ndash47 2009

[4] A Borumand Saeid ldquoVague BCKBCI-algebrasrdquo OpusculaMathematica vol 29 no 2 pp 177ndash186 2009

[5] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939

[6] T S Blyth and M F Janowitz Residuation Theory PergamonPress 1972

[7] T Kowalski and H Ono Residuated Lattices An AlgebraicGlimpse at Logic without Contraction Japan Advanced Instituteof Science and Technology March 2001

[8] D PiciuAlgebras of Fuzzy Logic Editura Universitaria CraiovaCraiova Romania 2007

[9] E TurunenMathematics Behind Fuzzy Logic Physica 1999[10] P Hajek Metamathematics of Fuzzy Logic vol 4 of Trends

in LogicmdashStudia Logica Library Kluwer Academic DordrechtThe Netherlands 1998

[11] Sh Ghorbani ldquoSub positive implicative filters of residuatedlatticesrdquo World Applied Science Journal vol 12 no 5 pp 586ndash590 2011

[12] S Ghorbani and A Hasankhani ldquoFuzzy convex subalgebrasof commutative residuated latticesrdquo Iranian Journal of FuzzySystems vol 7 no 2 pp 41ndash54 159 2010

[13] R Biswas ldquoVague groupsrdquo International Journal of Computa-tional Cognition vol 4 pp 20ndash23 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Discrete Mathematics 7

Proof Let 119860 119861 119862 isin VFD(119871) be such that 119860 ⪯ 119862 We willshow that (119860 lowast 119861) cap 119862 = (119860 lowast 119862) cap (119861 lowast 119862) By Lemma 32

119881(119860lowast119861)cap119862 (119911)

= imin (119881119860lowast119861

) (119911) 119881119862 (119911)

= imin isup imin 119881119860 (119909) 119881119861 (119910) 119911 ge 119909 lowast 119910 119881

119862 (119911)

= isup imin 119881119860 (119909) 119881119861 (119910) 119881

119862 (119911) 119911 ge 119909 lowast 119910

= isup imin 119881119860 (119909) 119881119861 (119910) 119881

119862(119909 lowast 119910) 119911 ge 119909 lowast 119910

= isup imin 119881119860 (119909) 119881119861 (119910) 119881

119862 (119909) 119881119862 (119910) 119911 ge 119909 lowast 119910

= isup imin 119881119860 (119909) 119881119862 (119909)

imin 119881119861(119910) 119881

119862(119910) 119911 ge 119909 lowast 119910

= isup imin 119881119860cap119862 (119909) 119881119861cap119862 (119910) 119911 ge 119909 lowast 119910

= 119881(119860cap119862)lowast(119861cap119862) (119911)

(29)

5 Subpositive Implicative Vague Filters

Definition 34 Let 119860 be a vague subset of a residuated lattice119871 119860 is called a subpositive implicative vague filter of 119871 if

(VF6) 119881119860(119909) le 119881

119860(1) for all 119909 isin 119871

(SVF1) imin119881119860(((119909 rarr 119910) lowast 119911) 999492999484 ((119910 999492999484 119909) rarr

119909)) 119881119860(119911) le 119881

119860((119909 rarr 119910) 999492999484 119910) for any

119909 119910 119911 isin 119871(SVF2) imin119881

119860((119911 lowast (119909 999492999484 119910)) rarr ((119910 rarr 119909) 999492999484

119909)) 119881119860(119911) le 119881

119860((119909 999492999484 119910) rarr 119910) for any

119909 119910 119911 isin 119871

Proposition 35 Let 119860 be a vague filter of a residuated lattice119871 Then 119860 is a subpositive implicative vague filter of L if andonly if it satisfies

(SVF3) 119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) le

119881119860((119909 rarr 119910) 999492999484 119910) for any 119909 119910 isin 119871

(SVF4) 119881119860((119909 999492999484 119910) rarr ((119910 rarr 119909) 999492999484 119909)) le

119881119860((119909 999492999484 119910) rarr 119910) for any 119909 119910 isin 119871

Proposition 36 Let119860 be a subpositive implicative vague filterof a residuated lattice 119871 Then 119860 is a vague filter of 119871

Proof We have imin119881119860(119909 999492999484 119910) 119881

119860(119909) = imin119881

119860((119910 rarr

119910) lowast 119909) 999492999484 ((119910 999492999484 119910) rarr 119910) 119881119860(119909) le 119881

119860((119910 rarr 119910) 999492999484

119910) = 119881119860(119910) By (VS1) andTheorem 17 119860 is a vague filter of a

residuated lattice 119871

The relation between subpositive implicative vague filterand its level subset is as follows

Theorem 37 Let 119860 be a vague set of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and only if

for all120572 120573 isin [0 1] the set119860(120572120573)

is either empty or a subpositiveimplicative filter of 119871 where 120572 le 120573

Proof Suppose that119860 is a subpositive implicative filter vagueset of 119871 and 120572 120573 isin [0 1] Let 119860

(120572120573)= 0 Since 119860 is a vague

filter then119860(120572120573)

is a filter of 119871 byTheorem 21 Now let (119909 rarr

119910) 999492999484 ((119910 999492999484 119909) rarr 119909) isin 119860(120572120573)

Then 119881119860((119909 rarr 119910) 999492999484

((119910 999492999484 119909) rarr 119909)) ge [120572 120573] By (SVF3) 119881119860((119909 rarr 119910) 999492999484

119910) ge [120572 120573] that is (119909 rarr 119910) 999492999484 119910 isin 119860(120572120573)

Similarly we canshow that119860

(120572120573)satisfies (SVF4) Hence119860

(120572120573)is a subpositive

implicative filter of 119871Conversely let 119905

119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) =

120572 and 1 minus 119891119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) = 120573 Thus

119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) ge [120572 120573] We get that

(119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909) isin 119860(120572120573)

So 119860(120572120573)

= 0Thus (119909 rarr 119910) 999492999484 119910 isin 119860

(120572120573) that is 119881

119860((119909 rarr 119910) 999492999484 119910) ge

[120572 120573] = 119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) Similarly we can

prove that 119860 satisfies (SVF4)

In the following theorems we obtain some characteriza-tions of subpositive implicative vague filter

Theorem 38 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF5) iminVA((y rarr z) 999492999484 (x rarr y))VA(x) le

VA(y) for any 119909 119910 119911 isin 119871(SVF6) iminVA((y 999492999484 z) rarr (x 999492999484 y))VA(x) le

VA(y) for any 119909 119910 119911 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871 ByProposition 2 part (3) we have imin119881

119860((119910 rarr 119911) 999492999484 (119909 rarr

119910)) 119881119860(119909) = imin119881

119860(119909 rarr ((119910 rarr 119911) 999492999484 119910)) 119881

119860(119909) le

119881119860((119910 rarr 119911) 999492999484 119910) By Proposition 2 part (7) (119910 rarr 119911) 999492999484

119910 le 119911 999492999484 119910 le (119910 rarr 119911) 999492999484 ((119911 999492999484 119910) rarr 119910) We get that119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119911 999492999484 119910) le 119881

119860((119910 rarr 119911) 999492999484 ((119911 999492999484

119910) rarr 119910)) le 119881119860((119911 999492999484 119910) rarr 119910) Hence imin119881

119860((119910 rarr

119911) 999492999484 119910) 119881119860(119911 999492999484 119910) le imin119881

119860((119911 999492999484 119910) rarr 119910) 119881

119860(119911 999492999484

119910) le 119881119860(119910) Thus it satisfies (SVF5) Similarly we can

prove (SVF6) Conversely let 119860 satisfy (SVF5) and (SVF6)119881119860((119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909)) = imin119881

119860([((119909 rarr 119910) 999492999484

119910) rarr 1] 999492999484 [(119909 rarr 119910) 999492999484 ((119910 999492999484 119909) rarr 119909) rarr ((119909 rarr

119910) 999492999484 119910)]) 119881119860(1) le 119881

119860((119909 rarr 119910) 999492999484 119910) Similarly we

can prove that (SVF5) holds Hence 119860 is a vague subpositiveimplicative filter

Theorem 39 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF7) 119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) le

119881119860((119910 999492999484 119909) rarr 119909) for any 119909 119910 isin 119871

(SVF8) 119881119860((119909 999492999484 119910) rarr ((119910 999492999484 119909) rarr 119909)) le

119881119860((119910 rarr 119909) 999492999484 119909) for any 119909 119910 isin 119871

Proof Let (SVF7) and (SVF8) hold We will show that 119860

satisfies (SVF5) and (SVF6) We have imin119881119860((119910 rarr 119911) 999492999484

(119909 rarr 119910)) 119881119860(119909) = imin119881

119860(119909 rarr ((119910 rarr 119911) 999492999484

119910)) 119881119860(119909) le 119881

119860((119910 rarr 119911) 999492999484 119910) le 119881

119860((119910 rarr 119911) 999492999484

8 Journal of Discrete Mathematics

((119911 rarr 119910) 999492999484 119910)) le 119881119860((119911 999492999484 119910) rarr 119910)) Also we have

119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119911 999492999484 119910) Since 119860 is a vague

filter 119881119860((119910 rarr 119911) 999492999484 119910) le imin119881

119860(119911 999492999484 119910) 119881

119860((119911 999492999484

119910) rarr 119910) le 119881119860(119910) Hence 119860 satisfies (SVF5) Analogously

we can prove that119860 satisfies (SVF6) Hence119860 is a subpositiveimplicative vague filter of 119871 byTheorem 38 Conversely let 119860be a subpositive implicative vague filter of 119871 We have

[((119910 999492999484 119909) 997888rarr 119909) 997888rarr 1]

999492999484 [((119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909))

997888rarr ((119910 999492999484 119909) 997888rarr 119909) ]

= [(119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)]

997888rarr [((119910 997888rarr 119909) 999492999484 119909) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)]

= (119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)

(30)

We get that 119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) =

imin119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909))119881

119860([((119910 999492999484 119909) rarr

119909) rarr 1] 999492999484 [((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) rarr ((119910 999492999484

119909) rarr 119909)]) le 119881119860((119910 999492999484 119909) rarr 119909) by (SVF5) Similarly we

can prove that (SVF8) holds

Theorem 40 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF9) 119881119860((119909 rarr 119910) 999492999484 119909) le 119881

119860(119909) for any 119909 119910 isin 119871

(SVF10) 119881119860((119909 999492999484 119910) rarr 119909) le 119881

119860(119909) for any 119909 119910 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871Then 119860 satisfies (SVF5) and (SVF6)

119881119860((119909 997888rarr 119910) 999492999484 119909)

= imin 119881119860(1 997888rarr ((119909 997888rarr 119910) 999492999484 119909)) 119881

119860 (1)

= imin 119881119860((119909 997888rarr 119910) 999492999484 (1 997888rarr 119909)) 119881119860 (1)

le 119881119860 (119909)

(31)

Similarly we can prove that (SVF10) holds Conversely let119860 satisfy (SVF9) and (SVF10) We will show that 119860 satisfies(SVF5) and (SVF6) imin119881

119860((119910 rarr 119911) 999492999484 (119909 rarr

119910)) 119881119860(119909) = imin119881

119860((119909 rarr (119910 rarr 119911) 999492999484 119910)) 119881

119860(119909) le

119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119910) Hence 119860 satisfies (SVF5)

Similarly we can prove (SVF7) holds Therefore 119860 is asubpositive implicative vague filter of L byTheorem 38

Theorem 41 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF11) 119881119860(not119909 999492999484 119909) le 119881

119860(119909) for any 119909 isin 119871

(SVF12) 119881119860(sim 119909 rarr 119909) le 119881

119860(119909) for any 119909 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871Then (SVF11) and (SVF12) follow by (SVF9) and (SVF10)Conversely since 0 le 119910 then 119909 rarr 0 le 119909 rarr 119910 Hence

(119909 rarr 119910) 999492999484 119909 le not119909 999492999484 119909 by Proposition 2 part (6)Therefore 119881

119860((119909 rarr 119910) 999492999484 119909) le 119881

119860(not119909 999492999484 119909) le 119881

119860(119909)

Similarly we can prove that (SVF10) holds Hence 119860 is asubpositive implicative vague filter of 119871 byTheorem 40

The extension theorem of subpositive implicative vaguefilters is obtained from the following theorem

Theorem 42 Let 119860 and 119861 be two vague filters of a residuatedlattice 119871 which satisfies 119860 le 119861 and 119881

119860(1) = 119881

119861(1) If 119860 is a

subpositive implicative vague filter of L so 119861 is

Proof Suppose that 119906 = (119909 rarr 119910) 999492999484 119909 Then

1 = 119906 997888rarr 119906 = 119906 997888rarr ((119909 997888rarr 119910) 999492999484 119909)

= (119909 997888rarr 119910) 999492999484 (119906 997888rarr 119909) = 1

(32)

We have (119909 rarr 119910) 999492999484 (119906 rarr 119909) le ((119906 rarr 119909) rarr 119910) 999492999484

(119906 rarr 119909) Therefore 119881119860(((119906 rarr 119909) rarr 119910) 999492999484 (119906 rarr 119909)) =

119881119860(1) By (SVF10) 119881

119860(1) = 119881

119860(((119906 rarr 119909) rarr 119910) 999492999484 (119906 rarr

119909)) le 119881119860(119906 rarr 119909) Hence 119881

119860(119906 rarr 119909) = 119881

119860(1) = 119881

119861(1)

Since 119860 le 119861 then 119881119861(119906 rarr 119909) ge 119881

119860(119906 rarr 119909) = 119881

119861(1)

We get that 119881119861((119909 rarr 119910) 999492999484 119909) = 119881

119861(119906) = imin119881

119861(119906 rarr

119909) 119881119861(119906) le 119881

119861(119909) Similarly we can show that 119866 satisfies

(SVF11) Hence 119866 is a vague subpositive implicative filter of119871 by Theorem 40

Definition 43 Let 119860 be a vague filter of a residuated lattice 119871119860 is called a vague Boolean filter of 119871 if

(BVF1) 119881119860(119909 or not119909) = 119881

119860(1) for all 119909 isin 119871

(BVF2) 119881119860(119909or sim 119909) = 119881

119860(1) for all 119909 isin 119871

Similar to Theorem 40 we have the following theoremwith respect to Boolean vague filters

Theorem 44 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a Boolean vague filter of 119871 if and only if for all120572 120573 isin [0 1] the set 119860

(120572120573)is either empty or a Boolean filter of

119871 where 120572 le 120573

Theorem 45 Let 119860 be a vague filter of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and if onlyif 119860 is a Boolean vague filter

Proof Let119860 be a subpositive implicative vague filterWe have

119881119860 (1) = 119881

119860 (not (119909 or not119909) 999492999484 (119909 or not119909))

= 119881119860 ((not119909 and notnot119909) 999492999484 (119909 or not119909)) le 119881

119860 (119909 or not119909)

(33)

Hence 119881119860(119909 or not119909) = 119881

119860(1) Similarly we can prove that 119860

satisfies (BVF2) Hence 119860 is a Boolean vague filterConversely let119881

119860(119909ornot119909) = 119881

119860(1)We have119881

119860(119909ornot119909) =

119881119860((119909 999492999484 119909) or (not119909 999492999484 119909))119881

119860((119909 or not119909) 999492999484 119909) = imin119881

119860((119909 or

not119909) 999492999484 119909) 119881119860(1) = imin119881

119860((119909 or not119909) 999492999484 119909) 119881

119860(119909 or not119909) le

119881119860(119909) Similarly we can show that119860 satisfies (SVF12) Hence

119860 is a subpositive implicative vague filter byTheorem 41

Journal of Discrete Mathematics 9

6 Conclusion

In this paper we introduced the notions of vague filterssubpositive implicative vague filters and Boolean vaguefilters of a residuated lattice and investigated some relatedproperties We studied the relationship between filters andvague filters of a residuated lattice and showed that vaguefilters of a residuated lattice is a generalization of fuzzy filtersWe proved that the set of all vague filters of a residuated latticeforms a complete lattice and obtained some characterizationsof subpositive implicative vague filterThe extension theoremof subpositive implicative vague filters was obtained Finallyit was proved that subpositive implicative vague filters areequivalent to Boolean vague filters

Conflict of Interests

There is no conflict of interests regarding the publication ofthis paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993

[3] Y B Jun and C H Park ldquoIdeals of pseudo MV-algebras basedon vague set theoryrdquo Iranian Journal of Fuzzy Systems vol 6 no7 pp 31ndash47 2009

[4] A Borumand Saeid ldquoVague BCKBCI-algebrasrdquo OpusculaMathematica vol 29 no 2 pp 177ndash186 2009

[5] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939

[6] T S Blyth and M F Janowitz Residuation Theory PergamonPress 1972

[7] T Kowalski and H Ono Residuated Lattices An AlgebraicGlimpse at Logic without Contraction Japan Advanced Instituteof Science and Technology March 2001

[8] D PiciuAlgebras of Fuzzy Logic Editura Universitaria CraiovaCraiova Romania 2007

[9] E TurunenMathematics Behind Fuzzy Logic Physica 1999[10] P Hajek Metamathematics of Fuzzy Logic vol 4 of Trends

in LogicmdashStudia Logica Library Kluwer Academic DordrechtThe Netherlands 1998

[11] Sh Ghorbani ldquoSub positive implicative filters of residuatedlatticesrdquo World Applied Science Journal vol 12 no 5 pp 586ndash590 2011

[12] S Ghorbani and A Hasankhani ldquoFuzzy convex subalgebrasof commutative residuated latticesrdquo Iranian Journal of FuzzySystems vol 7 no 2 pp 41ndash54 159 2010

[13] R Biswas ldquoVague groupsrdquo International Journal of Computa-tional Cognition vol 4 pp 20ndash23 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Discrete Mathematics

((119911 rarr 119910) 999492999484 119910)) le 119881119860((119911 999492999484 119910) rarr 119910)) Also we have

119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119911 999492999484 119910) Since 119860 is a vague

filter 119881119860((119910 rarr 119911) 999492999484 119910) le imin119881

119860(119911 999492999484 119910) 119881

119860((119911 999492999484

119910) rarr 119910) le 119881119860(119910) Hence 119860 satisfies (SVF5) Analogously

we can prove that119860 satisfies (SVF6) Hence119860 is a subpositiveimplicative vague filter of 119871 byTheorem 38 Conversely let 119860be a subpositive implicative vague filter of 119871 We have

[((119910 999492999484 119909) 997888rarr 119909) 997888rarr 1]

999492999484 [((119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909))

997888rarr ((119910 999492999484 119909) 997888rarr 119909) ]

= [(119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)]

997888rarr [((119910 997888rarr 119909) 999492999484 119909) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)]

= (119909 997888rarr 119910) 999492999484 ((119910 997888rarr 119909) 999492999484 119909)

(30)

We get that 119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) =

imin119881119860((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909))119881

119860([((119910 999492999484 119909) rarr

119909) rarr 1] 999492999484 [((119909 rarr 119910) 999492999484 ((119910 rarr 119909) 999492999484 119909)) rarr ((119910 999492999484

119909) rarr 119909)]) le 119881119860((119910 999492999484 119909) rarr 119909) by (SVF5) Similarly we

can prove that (SVF8) holds

Theorem 40 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF9) 119881119860((119909 rarr 119910) 999492999484 119909) le 119881

119860(119909) for any 119909 119910 isin 119871

(SVF10) 119881119860((119909 999492999484 119910) rarr 119909) le 119881

119860(119909) for any 119909 119910 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871Then 119860 satisfies (SVF5) and (SVF6)

119881119860((119909 997888rarr 119910) 999492999484 119909)

= imin 119881119860(1 997888rarr ((119909 997888rarr 119910) 999492999484 119909)) 119881

119860 (1)

= imin 119881119860((119909 997888rarr 119910) 999492999484 (1 997888rarr 119909)) 119881119860 (1)

le 119881119860 (119909)

(31)

Similarly we can prove that (SVF10) holds Conversely let119860 satisfy (SVF9) and (SVF10) We will show that 119860 satisfies(SVF5) and (SVF6) imin119881

119860((119910 rarr 119911) 999492999484 (119909 rarr

119910)) 119881119860(119909) = imin119881

119860((119909 rarr (119910 rarr 119911) 999492999484 119910)) 119881

119860(119909) le

119881119860((119910 rarr 119911) 999492999484 119910) le 119881

119860(119910) Hence 119860 satisfies (SVF5)

Similarly we can prove (SVF7) holds Therefore 119860 is asubpositive implicative vague filter of L byTheorem 38

Theorem 41 Let 119860 be a vague filter of a residuated lattice 119871Then 119860 is a subpositive implicative vague filter of 119871 if and onlyif it satisfies

(SVF11) 119881119860(not119909 999492999484 119909) le 119881

119860(119909) for any 119909 isin 119871

(SVF12) 119881119860(sim 119909 rarr 119909) le 119881

119860(119909) for any 119909 isin 119871

Proof Let 119860 be a subpositive implicative vague filter of 119871Then (SVF11) and (SVF12) follow by (SVF9) and (SVF10)Conversely since 0 le 119910 then 119909 rarr 0 le 119909 rarr 119910 Hence

(119909 rarr 119910) 999492999484 119909 le not119909 999492999484 119909 by Proposition 2 part (6)Therefore 119881

119860((119909 rarr 119910) 999492999484 119909) le 119881

119860(not119909 999492999484 119909) le 119881

119860(119909)

Similarly we can prove that (SVF10) holds Hence 119860 is asubpositive implicative vague filter of 119871 byTheorem 40

The extension theorem of subpositive implicative vaguefilters is obtained from the following theorem

Theorem 42 Let 119860 and 119861 be two vague filters of a residuatedlattice 119871 which satisfies 119860 le 119861 and 119881

119860(1) = 119881

119861(1) If 119860 is a

subpositive implicative vague filter of L so 119861 is

Proof Suppose that 119906 = (119909 rarr 119910) 999492999484 119909 Then

1 = 119906 997888rarr 119906 = 119906 997888rarr ((119909 997888rarr 119910) 999492999484 119909)

= (119909 997888rarr 119910) 999492999484 (119906 997888rarr 119909) = 1

(32)

We have (119909 rarr 119910) 999492999484 (119906 rarr 119909) le ((119906 rarr 119909) rarr 119910) 999492999484

(119906 rarr 119909) Therefore 119881119860(((119906 rarr 119909) rarr 119910) 999492999484 (119906 rarr 119909)) =

119881119860(1) By (SVF10) 119881

119860(1) = 119881

119860(((119906 rarr 119909) rarr 119910) 999492999484 (119906 rarr

119909)) le 119881119860(119906 rarr 119909) Hence 119881

119860(119906 rarr 119909) = 119881

119860(1) = 119881

119861(1)

Since 119860 le 119861 then 119881119861(119906 rarr 119909) ge 119881

119860(119906 rarr 119909) = 119881

119861(1)

We get that 119881119861((119909 rarr 119910) 999492999484 119909) = 119881

119861(119906) = imin119881

119861(119906 rarr

119909) 119881119861(119906) le 119881

119861(119909) Similarly we can show that 119866 satisfies

(SVF11) Hence 119866 is a vague subpositive implicative filter of119871 by Theorem 40

Definition 43 Let 119860 be a vague filter of a residuated lattice 119871119860 is called a vague Boolean filter of 119871 if

(BVF1) 119881119860(119909 or not119909) = 119881

119860(1) for all 119909 isin 119871

(BVF2) 119881119860(119909or sim 119909) = 119881

119860(1) for all 119909 isin 119871

Similar to Theorem 40 we have the following theoremwith respect to Boolean vague filters

Theorem 44 Let 119860 be a vague set of a residuated lattice 119871Then 119860 is a Boolean vague filter of 119871 if and only if for all120572 120573 isin [0 1] the set 119860

(120572120573)is either empty or a Boolean filter of

119871 where 120572 le 120573

Theorem 45 Let 119860 be a vague filter of a residuated lattice 119871Then119860 is a subpositive implicative vague filter of 119871 if and if onlyif 119860 is a Boolean vague filter

Proof Let119860 be a subpositive implicative vague filterWe have

119881119860 (1) = 119881

119860 (not (119909 or not119909) 999492999484 (119909 or not119909))

= 119881119860 ((not119909 and notnot119909) 999492999484 (119909 or not119909)) le 119881

119860 (119909 or not119909)

(33)

Hence 119881119860(119909 or not119909) = 119881

119860(1) Similarly we can prove that 119860

satisfies (BVF2) Hence 119860 is a Boolean vague filterConversely let119881

119860(119909ornot119909) = 119881

119860(1)We have119881

119860(119909ornot119909) =

119881119860((119909 999492999484 119909) or (not119909 999492999484 119909))119881

119860((119909 or not119909) 999492999484 119909) = imin119881

119860((119909 or

not119909) 999492999484 119909) 119881119860(1) = imin119881

119860((119909 or not119909) 999492999484 119909) 119881

119860(119909 or not119909) le

119881119860(119909) Similarly we can show that119860 satisfies (SVF12) Hence

119860 is a subpositive implicative vague filter byTheorem 41

Journal of Discrete Mathematics 9

6 Conclusion

In this paper we introduced the notions of vague filterssubpositive implicative vague filters and Boolean vaguefilters of a residuated lattice and investigated some relatedproperties We studied the relationship between filters andvague filters of a residuated lattice and showed that vaguefilters of a residuated lattice is a generalization of fuzzy filtersWe proved that the set of all vague filters of a residuated latticeforms a complete lattice and obtained some characterizationsof subpositive implicative vague filterThe extension theoremof subpositive implicative vague filters was obtained Finallyit was proved that subpositive implicative vague filters areequivalent to Boolean vague filters

Conflict of Interests

There is no conflict of interests regarding the publication ofthis paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993

[3] Y B Jun and C H Park ldquoIdeals of pseudo MV-algebras basedon vague set theoryrdquo Iranian Journal of Fuzzy Systems vol 6 no7 pp 31ndash47 2009

[4] A Borumand Saeid ldquoVague BCKBCI-algebrasrdquo OpusculaMathematica vol 29 no 2 pp 177ndash186 2009

[5] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939

[6] T S Blyth and M F Janowitz Residuation Theory PergamonPress 1972

[7] T Kowalski and H Ono Residuated Lattices An AlgebraicGlimpse at Logic without Contraction Japan Advanced Instituteof Science and Technology March 2001

[8] D PiciuAlgebras of Fuzzy Logic Editura Universitaria CraiovaCraiova Romania 2007

[9] E TurunenMathematics Behind Fuzzy Logic Physica 1999[10] P Hajek Metamathematics of Fuzzy Logic vol 4 of Trends

in LogicmdashStudia Logica Library Kluwer Academic DordrechtThe Netherlands 1998

[11] Sh Ghorbani ldquoSub positive implicative filters of residuatedlatticesrdquo World Applied Science Journal vol 12 no 5 pp 586ndash590 2011

[12] S Ghorbani and A Hasankhani ldquoFuzzy convex subalgebrasof commutative residuated latticesrdquo Iranian Journal of FuzzySystems vol 7 no 2 pp 41ndash54 159 2010

[13] R Biswas ldquoVague groupsrdquo International Journal of Computa-tional Cognition vol 4 pp 20ndash23 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Discrete Mathematics 9

6 Conclusion

In this paper we introduced the notions of vague filterssubpositive implicative vague filters and Boolean vaguefilters of a residuated lattice and investigated some relatedproperties We studied the relationship between filters andvague filters of a residuated lattice and showed that vaguefilters of a residuated lattice is a generalization of fuzzy filtersWe proved that the set of all vague filters of a residuated latticeforms a complete lattice and obtained some characterizationsof subpositive implicative vague filterThe extension theoremof subpositive implicative vague filters was obtained Finallyit was proved that subpositive implicative vague filters areequivalent to Boolean vague filters

Conflict of Interests

There is no conflict of interests regarding the publication ofthis paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993

[3] Y B Jun and C H Park ldquoIdeals of pseudo MV-algebras basedon vague set theoryrdquo Iranian Journal of Fuzzy Systems vol 6 no7 pp 31ndash47 2009

[4] A Borumand Saeid ldquoVague BCKBCI-algebrasrdquo OpusculaMathematica vol 29 no 2 pp 177ndash186 2009

[5] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939

[6] T S Blyth and M F Janowitz Residuation Theory PergamonPress 1972

[7] T Kowalski and H Ono Residuated Lattices An AlgebraicGlimpse at Logic without Contraction Japan Advanced Instituteof Science and Technology March 2001

[8] D PiciuAlgebras of Fuzzy Logic Editura Universitaria CraiovaCraiova Romania 2007

[9] E TurunenMathematics Behind Fuzzy Logic Physica 1999[10] P Hajek Metamathematics of Fuzzy Logic vol 4 of Trends

in LogicmdashStudia Logica Library Kluwer Academic DordrechtThe Netherlands 1998

[11] Sh Ghorbani ldquoSub positive implicative filters of residuatedlatticesrdquo World Applied Science Journal vol 12 no 5 pp 586ndash590 2011

[12] S Ghorbani and A Hasankhani ldquoFuzzy convex subalgebrasof commutative residuated latticesrdquo Iranian Journal of FuzzySystems vol 7 no 2 pp 41ndash54 159 2010

[13] R Biswas ldquoVague groupsrdquo International Journal of Computa-tional Cognition vol 4 pp 20ndash23 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of