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icant concern for the development and the applications of vague algebra. Formulation
of properties of these notions basically depends on a many-valued counterpart to the
tic are also given.
2005 Elsevier Inc. All rights reserved.
* Fax: +90 242 227 8911.
E-mail address: [email protected]
Information Sciences 176 (2006) 900936
www.elsevier.com/locate/insgeneralized associative law (called the generalized vague associative law in vague
groups). For this reason, the present paper and the forthcoming paper [M. Demirci,
The generalized associative law in vague groups and its applicationsII, Information
Sciences, Submitted for publication] are devoted to the formulation of the generalized
vague associative law and its applications in vague groups. The generalized vague asso-
ciative law in vague semigroups, which is the main contribution of this exposition, and
some elementary properties of the notion of a product of a nite number of elements in
vague semigroups, which cover necessary preparatory results for Part II, are the subjects
of this paper. Since the present paper forms an abstract foundation of the product and
sum of a nite number of real numbers in vague arithmetic, some practical applications
of the notions of product and sum of a nite number of real numbers in vague arithme-The generalized associative law invague groups and its applicationsI
Mustafa Demirci *
Department of Mathematics, Faculty of Sciences and Arts, Akdeniz University,
07058-Antalya, Turkey
Received 25 May 2004; received in revised form 7 October 2004; accepted 18 January 2005
Abstract
Products of elements and integral powers of elements in vague groups have a signif-0020-0255/$ - see front matter 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.ins.2005.01.020
natural implementation of the present results, it is shown in Section 3 how
the product and the sum of a nite number of real numbers can be explicitly
M. Demirci / Information Sciences 176 (2006) 900936 901calculated in vague arithmetic. Furthermore, some practical examples are de-
signed for this purpose. The generalized vague associative law will be stated
in Section 4, and it will be given in two main theorems (Theorems 4.4 andKeywords: Fuzzy groups; Vague groups; Fuzzy arithmetic; Vague arithmetic; Fuzzy equivalence
relations; Fuzzy equalities
1. Introduction
A new approach to the fuzzy settings of algebraic notions, diering from the
present literature of fuzzy algebra [27], has been introduced under the name
vague algebra in [8,10,11], and various elementary aspects of vague algebraicnotions have been studied in [8,10,11,28]. Vague algebra can be simply recog-
nized as a theory involving vaguely dened binary operations on nonempty sets
and their algebraic properties. Products of elements and integral powers of ele-
ments in the classical theory of groups have a signicant place in the develop-
ment of algebraic notions and the applications of algebraic notions to practical
considerations. In an analogy with classical algebra, notions of product and
integral power of elements in vague groups possess the same impact on pro-
gress in vague algebra. As pronounced in [16], the formulation of propertiesof these notions, especially properties of the latter notion, requires a vague
counterpart to the generalized associative law (called the generalized vague
associative law) in vague semigroups. Due to this fact, the present paper and
the follow-up paper [15] are devoted to the generalized vague associative law
and its applications to integral powers of elements in vague groups. The pur-
pose of this paper is to formulate the generalized vague associative law, and
to establish some elementary results related to products of elements in vague
semigroups, which will be needed in Part II. In order to display the signicanceand usefulness of the obtained results, we will also introduce some practical
applications of these results, and show how the present results can be imple-
mented practically. Integral powers of elements in vague groups and the appli-
cation of the generalized vague associative law to them will be subjects of
Part II.
The organization of the present paper can be expressed as follows: the
denition of vague semigroups and their relevant properties will be given
in the next section. Section 3 provides an overview to the notion of productof a nite number of elements in vague semigroups, and improves the rep-
resentation properties of vague products of elements presented in [16]. As a4.5).
902 M. Demirci / Information Sciences 176 (2006) 900936all L-fuzzy sets of X is denoted by LX. An L-fuzzy set q of X Y is called a fuzzyrelation from X to Y. In particular, if ~ is a fuzzy relation from X X to X,~x; y; z will be simply denoted by ~x; y; z for all x,y,z 2 X.
A map E :X X! L is called an M-equivalence relation on X [9] i thefollowing three conditions are satised:
(E.1) E(x,x) = 1, "x 2 X,(E.2) E(x,y) = E(y,x), "x,y 2 X,(E.3) E(x,y) * E(y,z) 6 E(x,z), "x,y,z 2 X.
An M-equivalence relation E on X is called an M-equality on X i E is
separated, i.e.
0mutative cqm-lattice [9,22])M = (L,6, *) provided with the following axioms:
(i) (L,6) is a complete lattice with the meet operation ^ on L, the join oper-ation _ on L, the bottom element 0 of L and the top element 1 of L, and05 1.
(ii) (L, *) is a commutative monoid with the identity 1.
(iii) * is isotone w.r.t. both rst and second arguments, i.e.
8a1; a2; b1; b2 2 L a1 6 a2 and b1 6 b2 ) a1 b1 6 a2 b2 :
Given an integral, commutative cqm-lattice M = (L,6, *), if there exists abinary operation! (called the residuum operation) on L satisfying the adjunc-tion property
AD a b 6 c() a 6 b! c; 8a; b; c 2 L;then M = (L,6, *) is called an integral, commutative cl-monoid [19].
In an integral, commutative cl-monoidM = (L,6, *), the monoid operation *is distributive over arbitrary joins [19], i.e. for each subset {biji 2 I} of L and foreach a 2 L, the equality a * (i2Ibi) =i2I(a * bi) holds. For a t-norm (left con-tinuous t-norm) *, it is well-known that the triple ([0,1],6,*) denes an integral,commutative cqm-lattice (an integral, commutative cl-monoid). In this paper,
M = (L,6, *) always stands for an integral, commutative cqm-lattice, unlessotherwise stated. The notation ni1ai will simply denote a1 * a2 * * an forn 2 N and a1,a2, . . .,an 2 L. The capital letters X and Y always represent non-empty usual sets. A function l :X! L is called an L-fuzzy set ofX, and the set of2. A brief introduction to vague semigroups
Before the introduction of vague semigroups, we rst recall the many-valued
logical base of this paper, which is dened by a xed triple (called an integral, com-(E.1 ) E(x,y) = 1 ) x = y, "x,y 2 X.
[4], a *-indistinguishability operator on X [29] and a similarity relation
Ex; y 6 F f x; f y; 8x 2 X ; 8y 2 Y :
M. Demirci / Information Sciences 176 (2006) 900936 903(ii) A fuzzy relation q 2 LXY is called a strong fuzzy function from X to Yw.r.t. E and F iff q satises the next conditions:(F.1) For each x 2 X, $y 2 Y such that q (x,y) = 1,(F.2) q(x,y) * q(x 0,y 0) * E(x,x 0) 6 F(y,y 0), "x,x 0 2 X, "y,y 0 2 Y.
(iii) A fuzzy relation q 2 LXY is called a perfect fuzzy function from X to Yw.r.t. E and F iff q fullls the condition (F.1) and the following conditions:(EX.1) q(x,y) * E(x,x 0) 6 q(x 0,y),(EX.2) q(x,y) * F(y,y 0) 6 q(x,y 0),(PF) q(x,y) * q(x,y 0) 6 F(y,y 0),
for all x,x 0 2 X and for all y,y 0 2 Y.
Denition 2.2 [10]. Let P and E be M-equivalence relations on X X and X,respectively.
(i) A strong fuzzy function (perfect fuzzy function) ~ from X X to X w.r.t. Pand E is said to be an M-vague binary operation (perfect M-vague binary"y 2 X, is called the extensional hull of x w.r.t. E.Vague binary operations form the essential tools of vague algebra and vague
arithmetic [8,10,11], and they are dened as particular strong fuzzy functions.
For this reason, it is useful to recall strong and perfect fuzzy functions, studied
in [79].
Denition 2.1. Let E and F be two M-equivalence relations on X and Y,
respectively.
(i) An ordinary function f :X! Y is said to be extensional w.r.t. E and F iffequality on X [18], a separated globalM-valued equality on X [20], a *-equality
[2,5] and a *-fuzzy equality on X [79,28].
AnM-equivalence relation E on X induces an equivalence relation E on X,dened by x E y () E(x,y) = 1, "x,y 2 X. Given an M-equivalence rela-tion E on X and x 2 X, the L-fuzzy set [x]E of X, dened by [x]E(y) = E(x,y),[1,14,30]. Furthermore, anM-equality on X is also known as a globalM-valuedIf we consider some special cases of the integral, commutative cqm-lattice
M = (L,6, *), an M-equivalence relation on X is also called an L-equality rela-tion [17], an M-valued similarity relation [18], a global M-valued equality
[20,21], an equality relation on X w.r.t. * [25], an L-fuzzy equivalence on Xw.r.t. * [24], a fuzzy equivalence relation on X w.r.t. * [3], a *-equivalenceoperation) on X w.r.t. P and E.
904 M. Demirci / Information Sciences 176 (2006) 900936Given an M-vague binary operation ~ on X w.r.t. P and E , an extensionalcrisp binary operation on X w.r.t. P and E, fullling the conditions (2.1) inTheorem 2.3, is called an ordinary description of ~, and the set of all ordinaryassumption ~ is transitive of the rst order. In other words, a vague binaryoperation is a perfect vague binary operation i it is the transitive of the rst
order (see [9, Remark 4.13(i)]).Theorem 2.3 shows that a perfect M-vague binary operation ~ is obviouslyan M-vague binary operation, and the converse will also be true under the(ii) AnM-vague binary operation ~ on X w.r.t. P and E is said to be transitiveof the rst order iff
(T.1) ~a; b; c Ec; d 6 ~a; b; d, "a,b,c,d 2 X.(iii) AnM-vague binary operation ~ on X w.r.t. P and E is said to be transitive
of the second order iff(T.2) ~a; b; c Eb; d 6 ~a; d; c, "a,b,c,d 2 X.
(iv) AnM-vague binary operation ~ on X w.r.t. P and E is said to be transitiveof the third order iff
(T.3) ~a; b; c Ea; d 6 ~d; b; c, "a,b,c,d 2 X.(v) X together with an M-vague binary operation (a perfect M-vague binary
operation), denoted by X ; ~, is called an M-vague semigroup (a perfectM-vague semigroup) w.r.t. P and E iff the following condition of vague
associativity is satised:
(VAS) ("a,b,c,d,m,q,w 2 X) ~b;c;d ~a;d;m ~a;b;q ~q;c;w6Em;w.
Vague binary operations and perfect vague binary operations can be repre-
sented by means of crisp binary operations:
Theorem 2.3. For a given M-vague binary operation ~ (resp. perfect M-vaguebinary operation) on X w.r.t. P and E, there exists at least one crisp binary
operation on X extensional w.r.t. P and E such that for all x,y, z 2 X,~x; y; x y 1 and ~x; y; z 6 Ex y; zresp: ~x; y; z Ex y; z: 2:1
Conversely, for a given crisp binary operation on X extensional w.r.t. P and E,an L-fuzzy relation ~ 2 LXX X , satisfying the conditions (resp. the equality) in(2.1), is an M-vague binary operation ~ (resp. perfect M-vague binary operation)on X w.r.t. P and E.
Proof. Cf. [9, Theorems 4.10 and 4.11]. hdescriptions of ~ is denoted by ORD~. If is a crisp binary operation on X
An M-equivalence relation E on X is said to right (left) regular w.r.t. a crisp
binary operation on X iEx; y 6 Ex u; y u Ex; y 6 Eu x; u y; 8x; u; y 2 X :
E is regular w.r.t. i it is both right and left regular w.r.t. [11]. Regularity ofM-equivalence relations play a signicant role for the constructions of vaguealgebraic notions in terms of the corresponding crisp algebraic notions
[8,11]. Some of their useful properties, which will be needed, are given in the
following proposition.
Proposition 2.4. Let X ; ~ be an M-vague semigroup w.r.t. P and E, and 2 ORD~. Then the following statements are valid:
(i) vag() is transitive of the third (second) order iff E is right (left) regularw.r.t. .
(ii) If E is regular w.r.t. , then (X,vag()) is a perfect M-vague semigroupw.r.t. P and E.
Proof. (i) Let vag() be transitive of the third order. Then consideringTheorem 2.3, we may write
Ex; y vagx; u; x u Ex; y 6 vagy; u; x u 6 Ey u; x u Ex u; y u; 8x; y; u 2 X ;
so the right regularity is obvious. Similarly, if vag() is transitive of the secondoperation on R w.r.t. ER2 and ER i the usual multiplication (addition) opera-tion (+) is an ordinary description of ~ (~) [1012]. If an M-vague mul-tiplication (addition) operation ~ (~) is a perfect M-vague binary operation onR, then it is called a perfect M-vague multiplication (addition) operation. Due
to Theorem 2.3, a perfectM-vague multiplication (addition) operation ~ (~) onR w.r.t. ER2 and ER is explicitly given by
~x; y; z ERx y; z ~x; y; z ERx y; z; 8x; y; z 2 X :extensional w.r.t. P and E, the perfect M-vague binary operation
vag() 2 L(XX)X, given by vag()(x,y,z) = E(x y,z), "x,y,z 2 X, is calledthe vague description of . If theM-equivalence relation E on X is anM-equal-ity, then ~ has a unique ordinary description, denoted by ord~ (see [9, Re-mark 4.13(ii)]).
For X R, an M-vague binary operation ~ (~) on R w.r.t. M-equivalencerelations ER2 on R
2 and ER on R is called anM-vague multiplication (addition)
M. Demirci / Information Sciences 176 (2006) 900936 905order, we have
vagx; y; z Ex; x Ex y; z Ex; x 6 Ex y; z Ex y; x y6 Ex0 y; z vagx0; y; z; 8x; x0; y; z 2 X ;i.e. vag() is transitive of the third order. In a similar fashion to the right reg-ularity of E w.r.t. , if E is assumed to be left regular w.r.t. , we see that
vagx; y; z Ey; y0 Ex y; z Ey; y06 Ex y; z Ex y; x y0 6 Ex y 0; z vagx; y0; z; 8x; y; y0; z 2 X ;
i.e. vag() is transitive of the second order.(ii) Assume that E is regular w.r.t. . From [10, Theorem 2.10 (v)], we have
Ea b c; a b c 1:Then using the regularity of E w.r.t. , and by Theorem 2.3, we observe that
vagb; c; d vaga; d;m vaga; b; q vagq; c;w Eb c; d Ea d;m Ea b; q Eq c;w6 Ea b c; a d Ea d;m Ea b c; q c Eq c;w6 Ea b c;m Ea b c;w Ea b c;m Ea b c;w Ea b c; a b c6 Em;w;
so vag() fullls the condition (VAS) of vague associativity, i.e. (X, vag()) is aperfect M-vague semigroup w.r.t. P and E. h
3. Products of elements in vague groups and their fundamental properties
The notion of the vague product of a nite number of elements in vague
semigroups and its fundamental aspects are introduced in [16]. One of the aims
of this section is to recall this notion and some of its elementary properties,
which will be needed in this paper and the companion paper [15]. The otherEx; y vagu; x; u x Ex; y 6 vagu; y; u x 6 Eu y; u x Eu x; u y; 8x; y; u 2 X ;
i.e. E is left regular w.r.t. . Conversely, suppose that E is right regular w.r.t. .Then
0 0 0
906 M. Demirci / Information Sciences 176 (2006) 900936aim is to give some practical applications of these results, and to show how
these results can be realized in practical considerations. This subject will be
1 2 n
act right product of a ,a , . . .,a 2 X coincides with the exact left product of1 2 na1,a2, . . .,an 2 X in the vague semigroup X ; ~, and it will be nothing but theproduct of a1,a2, . . .,an in (X,) in the usual sense. If E is not an M-equalityon X, Qni1aiRfi ji1;...;n1g does not necessarily coincide withQni1aiLfi ji1;...;n1g, but they relate to each other by the following rule:Proposition 3.2 [16]. For n 2 N, for all a1,a2, . . ., an 2 X and for all{iji = 1, . . ., n1}, fDiji 1; . . . ; n 1g ORD~, Qni1aiRfDiji1;...;n1g,Qn L Qn R Qn Li1 i fiji1;...;n1g i1 i fi ji1;...;n1gthe exact product of the elements a1,a2, . . .,an 2 X w.r.t. {iji =1, . . .,n1}, and is denoted by Qni1aifi ji1;...;n1g.
In Denition 3.1, if fi;Dji 1; . . . ; n 1g ORD~ and i = D forall i = 1, . . .,n 1, then Qni1aifiji1;...;n1g (resp., Qni1aiRfiji1;...;n1g,Qni1aiLfi ji1;...;n1g) will be denoted by Qni1aiD (resp., Qni1aiRD , Qni1aiLD ).If the M-equivalence relation E in Denition 3.1 is particularly taken as an
M-equality on X, then ~ has a unique ordinary description , and by [10,Theorem 2.10(vi)], (X,) forms a semigroup in the classical sense. Thus,for all fiji 1; . . . ; n 1g ORD~, we have that Qni1aiRfiji1;...;n1g Qni1aiLfi ji1;...;n1g Qni1ai, and Qni1ai is exactly the product ofa ,a , . . .,a 2 X in the semigroup (X,) in the classical sense. Therefore, the ex-nite number of elements, in vague semigroups [16]. For this reason, we rst
need to review these subsidiary tools. In the rest of this paper, we always as-
sume that X ; ~ is an M-vague semigroup w.r.t. two arbitrarily xed M-equi-valence relations P on X X and E on X , and ~ denotes a multiplicativenotation, unless otherwise mentioned.
Denition 3.1 [16]. Let fiji 1; . . . ;n 1g ORD~ and a,a1,a2, . . .,an 2 Xfor nP 2.
(i) The element ( ((a1 1 a2) 2 a3) ) n1 an (a1 n1 (a2 n2 (a3 n3( (an2 2 (an1 1 an)) )))) of X, denoted by
Qni1aiRfi ji1;...;n1g
(Qni1aiLfi ji1;...;n1g), is called the exact right (left) product of the elementsa1,a2, . . .,an 2 X w.r.t. {iji = 1, . . .,n 1}.
(ii) For n = 1, both Qni1aiRfi ji1;...;n1g and Qni1aiLfi ji1;...;n1g are dened tobe a1.
(iii) If Qn a R coincides with Qn a L , then it is calledhandled towards the end of this section. The formulation of the considered no-
tion and its properties basically depend on two subsidiary mathematical tools,
exact product of a nite number of elements and successive product of a
M. Demirci / Information Sciences 176 (2006) 900936 907 i1aifiji1;...;n1g, i1aifiji1;...;n1g and i1aifDiji1;...;n1g belong to thesame equivalence class according to E.
W~ a1; a2; . . . ; an; u1; . . . ; un i1 ;
W~ a1; a2; . . . ; an; u1; . . . ; un 0; if u 6 a: ;;(i) WR~ a1; a2; . . . ; an; u1; . . . ; un (WL~ a1; a2; . . . ; an; u1; . . . ; un) is saidto be the degree for which the n-tuple (u1, . . .,un) 2 Xn is a right (left) suc-cessive product of a1,a2, . . .,an 2 X.
(ii) An n-tuple (u1, . . .,un) 2 Xn is said to be an exact right (left) successiveproduct of a1,a2, . . .,an 2 X iff u1 = a1 (un = an) and WR~ a1; a2; . . . ;an; u1; . . . ; un 1 (WL~ a1; a2; . . . ; an; u1; . . . ; un 1).
Successive products in both the graded sense and exact sense are also two
subsidiary tools of vague products other than exact products. But the exact
successive product and the exact product are not independent notions, and
they are associated with each other in the following manner:
Proposition 3.4 [16]. (u1, . . ., un) is an exact right (left) successive product ofa1,a2, . . ., an 2 X (nP 2) iff there exists fiji 1; . . . ; n 1g ORD~ suchthat
uk Yki1
ai
" #Rfi ji1;...;k1g
uk Ynk1i1
ak1i
" #Lfi ji1;...;nkg
0@ 1A;for all k = 1, . . ., n.
An analogous result to Proposition 3.4 can be established between gradedn n
for all a1,a2, . . .,an,u1, . . .,un 2 X and for nP 2, and
WR~ a; u WL~ a; u 1; if u a0; if u 6 a
( );
for all a,u 2 X and for n = 1.0; if u1 6 a1: ;L
n1i1
~ai; ui1; ui; ; if un an8< 9=Denition 3.3 [16]. Let us dene the maps WR~ ;WL~ : X
n Xn ! L by
R n1
~ui; ai1; ui1; if u1 a18< 9=
908 M. Demirci / Information Sciences 176 (2006) 900936successive products and exact products: for nP 2, a1,a2, . . .,an, x 2 X andfiji 1; . . . ; n 2g ORD~,
f ji 1; . . . ; n 2g ORD~; ePLa1; a2; . . . ; anx _ ~ a1; Yn1
i1ai1
" #Lfi ji1;...;n2g
; x
0@ 1A8
Taking into account the equalities (3.1) and (3.2), right (left) vague products
can be expressed in terms of graded right (left) successive products: for nP 2,a1,a2, . . .,an,x 2 X and fiji 1; . . . ; n 2g ORD~,ePRa1; a2; . . . ; anh ix
_
WRa1; a2; . . . ; an; a1; v2; v3; . . . ; vn1; xjvk
8
belonging to ORD~, then Proposition 3.6(iii) simply states that the rightvague product ePRa1; a2; . . . ; an coincides with the left vague productePLa1; a2; . . . ; an, and the common vague product ePa1; a2; . . . ; an is justthe extensional hull of the exact product Qni1aiD w.r.t. E. It should be notedhere that Qni1aiD is nothing but the product of a1,a2, . . .,an 2 X in the semi-group (X,D) in the classical sense.
The relations between the exact right products and the exact left products
are stated in Proposition 3.2. An analogous result to Proposition 3.2 can be ex-
pressed for the right vague products and the left vague products as follows:
Proposition 3.7. For all x,y,a1,a2, . . ., an 2 X, n 2 N and for U;V 2 fL;Rg,the relation
ePUa1; a2; . . . ; anx ePVa1; a2; . . . ; any 6 Ex; y:holds for all x,y 2 X.ePRa1; a2; . . . ; anh ix 6 E Yni1
ai
" #Ufiji1;...;n1g
; x
0@ 1Aand ePLa1; a2; . . . ; anh ix 6 E Yn
i1ai
" #Ufiji1;...;n1g
; x
0@ 1Afor all x 2 X. Furthermore the equalities in both of these two inequalitiesare satisfied if ~ is a perfect M-vague binary operation.
(iii) If ~ is a perfect M-vague binary operation provided that ~ has an associativeordinary description D 2 ORD~, then the right vague productePRa1; a2; . . . ; an is exactly the same as the left vague productePLa1; a2; . . . ; an, and the vague product ePa1; a2; . . . ; an is given byePa1; a2; . . . ; anh ix E Yn
i1ai
" #D
; x
!; 8x 2 X : 3:5
Proof. Cf. [16, Theorem 4.1, Corollary 4.2 and Theorem 4.7]. h
The second part of Proposition 3.6(ii) shows that for U;V 2 fR;Lg, thevague (right or left) product ePUa1; a2; . . . ; an is nothing but the extensionalhull of the exact (right or left) product Qni1aiVfiji1;...;n1g w.r.t. E, whenever~ is a perfect M-vague binary operation. Furthermore, if the perfect M-vaguebinary operation ~ is enriched with an associative crisp binary operation D
M. Demirci / Information Sciences 176 (2006) 900936 911Proof. For n = 1, the assertion is trivial. For nP 2, let us pick fiji 1; . . . ; n 1g ORD~. By Proposition 3.6(ii), we may write
ePa1; a2; . . . ; an is given by (3.5).
(ii) Consider a regular M-equivalence relation ER on R w.r.t. the usual addi-
tion (multiplication) operation + (). For an M-equivalence relationER2 on R
2, let ~ be a perfect M-vague addition (multiplication) operationon R w.r.t. ER2 and ER. Using [11, Corollary 2.15], it is not difcult toconclude that R; ~ denes a perfect M-vague semigroup. In additionto this, 2 ORD~ ( 2 ORD~) is obviously an associative ordinarydescription of ~. Thus, because of Proposition 3.6(iii), for nP 2, thevague sum eRa1; a2; . . . ; an (product ePa1; a2; . . . ; an) of given real num-bers a1; a2; . . . ; an 2 R w.r.t. the perfectM-vague addition (multiplication)operation ~ is simply calculated as:
eRa1; a2; . . . ; anh ix ER Xni1
ai; x
!
eh i Yn ! !ePUa1; a2; . . . ; anh ix 6 E Yni1
ai
" #Ufiji1;...;n1g
; x
0@ 1A andePVa1; a2; . . . ; anh iy 6 E Yn
i1ai
" #Ufiji1;...;n1g
; y
0@ 1A:Thus ePUa1; a2; . . . ; anh ix ePVa1; a2; . . . ; anh iy
6 EYni1
ai
" #Ufi ji1;...;n1g
; x
0@ 1A E Yni1
ai
" #Ufi ji1;...;n1g
; y
0@ 1A6 Ex; y:
Proposition 3.7 can be viewed as the many-valued logical representation of
the sentence: if x and y are, respectively, the (right or left) products ofa1,a2, . . .,an, then they are equal to each other.
Remark 3.8 [16].
(i) Suppose that E is an M-equality on X and X ; ~ is a perfect M-vaguesemigroup w.r.t. P and E. Then ~ has a unique ordinary descriptionD ord~ , and it follows from [10, Theorem 2.10(vi)] that (X,D) formsa semigroup. Thus, by Proposition 3.6(iii), for nP 2, the vague product
912 M. Demirci / Information Sciences 176 (2006) 900936Pa1; a2; . . . ; an x ERi1
ai; x ; 8x 2 R:
Example 3.9. (i) Denoting the Lukasiewiczs t-norm by Lck, i.e.Lck(x,y) = max{x + y 1,0}, "x,y 2 [0, 1], let us particularly choose theunderlying integral, commutative cqm-lattice as M = ([0,1],6,Lck). The mapsER : R R! 0; 1, ER2 : R2 R2 ! 0; 1 and the fuzzy relation ~ER 20; 1R3 , given by
ER x; y 1minfjx yj; 1g;
ER2x; y; x0; y0 ER x y; x0 y0
and
~ER
x; y; z ER x y; z; 8x; y; z; x0; y0 2 R;are, respectively, M-equivalence relations (or simply Lck-indistinguishability
operators) on R and R2 and a perfect M-vague addition operation on R
w.r.t. ER2
and ER (see [11, Example 4.5 (i)]). Because of the fact that ER is
regular w.r.t. +, it is an immediate consequence of Remark 3.8(ii) that fornP 2 the vague sum eRa1; a2; . . . ; an of given real numbers a1; a2; . . . ; an 2 Rw.r.t. the perfect M-vague addition operation ~
ER
is simply given by
eRa1; a2; . . . ; anh ix ER Xni1
ai; x
! 1min
Xni1
ai x
; 1( )
;
8x 2 R [16]:(ii) For the product t-norm Pr dened by Pr(x,y) = x y, let the underlying
integral, commutative cqm-lattice be M = ([0,1],6,Pr). The mapsER : R R! 0; 1, ER2 : R2 R2 ! 0; 1, dened by
ERx; y min xy
; yx n o; if x; y 2 R f0gEcRx; y; otherwise
8
monotonic map / : R! R converting c to k, which is known as the permissible
transformation [26] or simply a scale itself [6,23], if we denote two indistinguish-
ability operators on R by EkR and EcR that are considered in the measurement
according to the scales k and c, respectively, then EcR should relate to EkR by
the equality EcRx; y EkR/x;/y, 8x; y 2 R (see [13] for details).In the measurement process of the quantity q according to the scale k, for
kformulate the vague product ePa1; a2; . . . ; an of given real numbersa1; a2; . . . ; an 2 R w.r.t. the perfect M-vague multiplication operation ~ER by
ePa1; a2; . . . ; anh ix ER Yni1
ai; x
!; 8x 2 R:
Because of the fact that all measurement instruments involve discrete read-
ings (or scales), it is not possible to determine the value of a quantity q with in-
nite precision. Discereteness of scales of measurement instruments are a naturalsource of the uncertainty on the indistinguishability of any two possible values
of the considered quantity q. Handling this uncertainty within the framework of
two-valued logic (true-false logic) does not provide consistent decisions on
whether two possible values of q are certainly distinguishable from each other.
For example, we may deal with the measurement of the heights of two tables,
which have apparently the same height, by using two dierent rods, each having
the unit of centimeter (cm) and millimeter (mm), and so we may wish to decide
whether these two tables in their heights are certainly identical or indistinguish-able. Say that 100 cm is measured by the rod with the unit of cm for both tables,
and 998 and 1002 mm heights are measured by the other rod. This means that
the heights of these two tables are indistinguishable according to the rst mea-
surement, and are distinguishable according to the second measurement. These
two inconsistent decisions result from the assumption that indistinguishability is
a two-valued logical concept. In order to describe the notion of indistinguish-
ability in a consistent way, we basically assume that indistinguishability is not
a two-valued logical concept, but a many-valued logical one. In accordance withthis assumption, for M = ([0, 1],6, *) and for a suitably chosen M-equivalencerelation (or simply, an indistinguishability operator [29]) EkR on R, where k rep-resents the scale of the measurement instrument, we will grade the indistinguish-
ability of any two possible values x; y 2 R of the quantity q as the real numberEkRx; y in [0,1], where the case EkRx; y 1 (EkRx; y 0) can be interpreted sothat x and y are completely indistinguishable (distinguishable) from each other.
In the measurement process of two tables, since their heights are the properties
that are not aected by the selected measurement instruments or their scales, it isnatural to expect that indistinguishability of these properties should be indepen-
dent from the scales. Because of this fact, for two given scales k and c and for the
914 M. Demirci / Information Sciences 176 (2006) 900936suitably chosen indistinguishability operators ER : R R! 0; 1 on R andEkR2
: R2 R2 ! 0; 1 on R2 and for measured values or possible values
x; x0; y; y0 2 R of q, the degree of indistinguishability of x; y 2 R and the degreeof indistinguishability of ordered pairs (x,y) and (x 0,y 0) according to the scale kare given by EkRx; y and EkR2x; y; x0; y0 , respectively. Because of the con-sideration of indistinguishability operators EkR and E
kR2, it is natural to expect
vaguely dened addition and multiplication operations on R which are mod-
elled by anM-vague addition operation ~k and anM-vague addition operation~k w.r.t. Ek
R2and EkR. For the sake of simplicity, suppose that ~k (~k) is a perfect
M-vague addition (multiplication) operation on R w.r.t. EkR2
and EkR. Thenfor any possible values a1; a2; . . . ; an 2 R of the quantity q according to thescale k (nP 2), if we would like to calculate the sum (product) of these values,then the result will not be the real number
Pni1ai
Qni1ai, but a fuzzy quan-
tity, or simply the vague sum (product) eRka1; a2; . . . ; an ( ePka1; a2; . . . ; an) ofa1,a2, . . .,an w.r.t. ~k~k, formulated by:
eRka1; a2; . . . ; anh ix EkR Xni1
ai; x
!
ePka1; a2; . . . ; anh ix EkR Yni1
ai; x
! !; 8x 2 R:
Whenever another scale c is considered instead of k, it is interesting to deter-mine what kinds of scales (or their permissible transformations) allow us
to convert given indistinguishability operators EkR and EkR2
and the perfect
M-vague addition (multiplication) operation ~k~k on R to indistinguisha-bility operators EcR and E
cR2
and a perfect M-vague addition (multiplication)
operation ~c~c on R according to the scale c, respectively. As an importantconsequence of the solution to this problem, we can calculate the vague sum
(product) eRcb1; b2; . . . ; bn ( ePcb1; b2; . . . ; bn) of possible values b1; b2; . . . ;bn 2 R of the quantity q obtained in the measurement process according tothe scale c. In the example given below, we demonstrate how this calculationcan be done.
Example 3.10. (i) Let k and c be two interval scales [26] , i.e. for some a; b 2 Rwith a > 0, the afne transformation /int(x) = a x + b converts c to k (see
[26]). Let EkR, EkR2
and ~k be, respectively, the Lck-indistinguishabilityoperators ER and E
R2
and the perfect M-vague addition operation on R
w.r.t. ER2
and ER dened in Example 3.9(i). For the scale c, if we dene themaps EcR : R R! 0; 1, EcR2 : R2 R2 ! 0; 1 and the fuzzy relation ~
c 20; 1R3 by
M. Demirci / Information Sciences 176 (2006) 900936 915EcRx; y EkR/intx;/inty 1minfa jx yj; 1g;
EcR2x; y; x0; y 0 Ek
R2/intx;/inty; /intx0;/inty0
1minfa jx y x0 y 0j; 1g and~cx; y; z EcRx y; z
1minfa jx y zj; 1gfor all x; x0; y; y 0; z 2 R, then EcR and EcR2 are obviously Lck-indistinguishabilityoperators on R and R2, respectively. Furthermore, since the usual addition
operation + is extensional w.r.t. EcR and EcR2, it is evident from Theorem
2.3 that ~c is a perfect M-vague addition operation on R w.r.t. EcR2
and EcR.Therefore, because of the fact that EcR is regular w.r.t. +, we obtain from Re-mark 3.8(ii) that the vague sum eRcb1; b2; . . . ; bn of any possible valuesb1; b2; . . . ; bn 2 R of the quantity q (nP 2) w.r.t. ~c is calculated as:
eRcb1; b2; . . . ; bnx EcR Xni1
bi; x
!
1min a Xni1
bi x
; 1( )
; 8x 2 R:
(ii) For two log-interval scales k and c [26] and for some a;x 2 R such thata > 0 and x > 0 , let their permissible transformation (power transformation[26]) /logint(x) = a x
x be dened on R and convert c to k. Let us take EkR,EkR2
and ~k as the Pr-indistinguishability operators ER, ER2 and the perfectM-vague multiplication operation ~ER w.r.t. ER2 and ER dened in Example3.9(ii), respectively. Consider the maps EcR : R R! 0; 1, EcR2 : R2 R2 !0; 1 and the fuzzy relation ~c 2 0; 1R3 , dened by
EcRx; y EkR/log -intx;/log -inty
minxy
x ; yx x n o; if x; y 2 R f0gEcRx; y; otherwise
8>9>>=>>
916 M. Demirci / Information Sciences 176 (2006) 9009360; otherwise: ;
for all x; y; x0; y0 2 R, where EcR2
stands for the classical Pr-indistinguishability
operator on R2, i.e.
EcR2x; y; x0; y 0 1; if x; y x
0; y00; if x; y 6 x0; y0
:
Then EcR and EcR2
are obviously Pr-indistinguishability operators on R and R2,
respectively. Because of the fact that the usual product operation is anextensional function w.r.t. Ec
R2and EcR, Theorem 2.3 gives that ~c is a perfect
M-vague multiplication operation on R w.r.t. EcR2
and EcR. Therewith, usingc
M. Demirci / Information Sciences 176 (2006) 900936 917the fact that ER is regular w.r.t. , and by Remark 3.8(ii), the vague productePcb1; b2; . . . ; bn of any possible values b1; b2; . . . ; bn 2 R of the quantityq (nP 2) w.r.t. ~c is calculated as:
ePcb1; b2; . . . ; bnh ix EcR Yni1
bi; x
!
minQn
i1bix
x ; xQni1bi
x ; if x;Qni1bi 2 R f0gEcR
Qni1bi; x
; otherwise
8>:9>=>;
for all x 2 R.
As an example of a realization of Example 3.10(i), let us deal with the mea-
surement of the perimeter of a triangle A. In order to measure the length of its
sides, we begin with a rod having cm readings as our measurement instrument.
Assume that a1, a2 and a3 are the lengths of sides of A obtained in the measure-
ment process. The rod having cm readings represents here the scale k. If weoperate with the Lck-indistinguishability operators EkR, E
kR2
and the perfect
M-vague addition operation ~k in Example 3.10(i), then the perimeter of thetriangle A in the unit of cm will be the vague sum eRka1; a2; a3 of a1, a2 anda3 w.r.t. ~k, which is a triangular fuzzy number centered at a1 + a2 + a3 (seeFig. 1), given byFig. 1. The perimeter of the triangle A in the unit of cm.
eRka1; a2; a3h ix EkR X3i1
ai; x
! 1min
X3i1
ai x
; 1( )
; 8x 2 R:
To obtain more sensitive results, if we use another rod having mm readings
corresponding to the another scale c, then the lengths of sides of the triangle Awill be measured as b1, b2 and b3 according to this new rod (or simply the scale
c). The permissible transformation converting mm readings to cm readings ishere a similarity transformation [26], given by /ratiox x10, 8x 2 R, and so k
918 M. Demirci / Information Sciences 176 (2006) 900936and c are ratio scales [26]. Then taking into consideration the Lck-indistin-guishability operators EcR, E
cR2
and the perfect M-vague addition operation~c in Example 3.10(i), we now calculate the perimeter of the triangle A inthe unit of mm as the vague sum eRcb1; b2; b3 of b1, b2 and b3 w.r.t. ~c, givenby
eRcb1;b2;b3x EcR X3i1
bi;x
! 1min 1
10X3i1
bi x
;1( )
; 8x2R:
Here eRcb1; b2; b3 is a triangular fuzzy number centered at b1 + b2 + b3, and issketched in Fig. 2.
To give an example for practical applications of Example 3.10(ii), we mayconsider the measurement of volume of a right rectangular prism. For the pur-
pose of measurement of its sides, let us reconsider the same rods, correspond-
ing to ratio scales k and c, and the similarity transformation /ratiox x10between them in the preceding example. Let a1, a2 and a3 be the measured val-
ues of its lengths of sides according to k (c). Consider the Pr-indistinguishabil-ity operators EkR EcR, EkR2 EcR2 and the perfect M-vague multiplicationoperation ~k~c in Example 3.10(ii) for the side measurements according to k(c). Then the required volume in the units of cm3 and mm3 are, respectively,the vague products ePka1; a2; a3 and ePca1; a2; a3 of a1, a2 and a3 w.r.t. ~kand ~c, and they coincide with each other. The graph of ePka1; a2; a3 is illus-trated in Fig. 3, and is explicitly given byFig. 2. The perimeter of the triangle A in the unit of mm.
ePka1; a2; a3h ix ePca1; a2; a3h ix EkRa1 a2 a3; x min
a1a2a3jxj ;
jxja1a2a3
n o; if x 6 0
0; if x 0
( ):
As can easily be observed from these two practical applications of Example
3.10, if the scales k and c in Example 3.10(i) and (ii) are particularly taken asratio scales, i.e. they have the similarity transformation /ratio : R! R, denedby /ratio(x) = a x for some a > 0, as their permissible transformation [26], thenthe perfect M-vague addition (multiplication) operations ~k and ~c (~k and ~c)and the vague sums (products) eRk and eRc ( ePk and ePc) satisfy the followingequalities:
~cx; y; z ~k/ratiox;/ratioy;/ratioz and eRcb1; b2; . . . ; bnh ix eRk/ratiob1;/ratiob2; . . . ;/ratiobnh i/ratiox
~cx; y; z ~k/ratiox;/ratioy;/ratioz and ePcb1; b2; . . . ; bnh ix
Fig. 3. The volume of the right rectangular prism in the units of cm3 or mm3.M. Demirci / Information Sciences 176 (2006) 900936 919 ePk/ratiob1;/ratiob2; . . . ;/ratiobnh i/ratiox:These equalities can be interpreted as the invariance of the considered vagueaddition (multiplication) operations and the vague sums (products) of a nite
number of possible values of q under the ratio scales.
4. The generalized associative law in vague semigroups
For n, m 2 N and for given elements a1,a2, . . .,an+m of a semigroup (X,) inthe classical sense, the equality
Qnmi1 ai
Qni1ai
Qmi1ani is known as the
generalized associative law in classical algebra. The aim of this section is
to establish a vague counterpart to the generalized associative law for vague
1 nm 1 nm 1 n 1 n
6 WLan1; . . . ; anm; vn1; . . . ; vnm EunDvn1; v1.
In addition to the necessity of Lemmas 4.1 and 4.2 in the formulation of the
generalized associative law for vague products, they enable us to establish the
generalized associative law for exact products, which will also be required in
the sequel:
Proposition 4.3. Let X ; ~ be an M-vague semigroup w.r.t. P and E. Then for m,n 2 N, a1, . . ., an+m 2 X, D 2 ORD~ and {iji = 1, . . .,m 1},fDiji 1; . . . ; n m 1g ORD~, we have(i) Qni1aiRfDiji1;...;n1gDQmi1aniLfiji1;...;m1gEQnmi1 aiRfDi ji1;...;nm1g.Furthermore, if one of the conditions (a) or (b) in Lemma 4.1 is satisfied, then
the next properties are valid:
(ii) Qni1aiRfDiji1;...;n1gDQmi1aniRfiji1;...;m1gEQnmi1 aiRfDi ji1;...;nm1g.anm;un1; . . . ;unmEv1Dun1;u1.
Lemma 4.2. For m; n 2 N, a1,a2, . . ., an+m,u1, . . ., un+m, v1,v2, . . ., vn+m 2 X andD 2 ORD~ , the following inequalities are satisfied:
(i) WRa1; . . . ; anm; u1; . . . ; unm WLan1; . . . ; anm;v1; . . . ; vm 6 WRa1; . . . ; an; u1; . . . ; un EunDv1; unm.
(ii) WLa ; . . . ; a ; v ; . . . ; v WRa ; . . . ; a ; u ; . . . ; u (a) ~ is transitive of the first order, or equivalently ~ is perfect.(b) E is regular w.r.t. D.
Then for m; n 2 N, a1,a2, . . ., an+m,u1, . . ., un+m, v1, v2, . . ., vn+m 2 X, the fol-lowing properties are true:
(i) WRa1; . . . ;anm;u1; . . . ;unmWRan1; . . . ;anm;v1; . . . ;vm6EunDvm;unm.
(ii) WLa1; . . . ;anm;u1; . . . ;unmWLa1; . . . ;an;v1; . . . ;vn6WLan1; . . . ;products. For this purpose, we rst need to formulate the generalized associa-
tive law for graded successive products, presented in the subsequent lemmas:
Lemma 4.1. For a given D 2 ORD~, let one of the following two conditions besatisfied:
920 M. Demirci / Information Sciences 176 (2006) 900936(iii) Qni1aiLfDiji1;...;n1gDQmi1aniLfiji1;...;m1gEQnmi1 aiLfDi ji1;...;nm1g.
Theorem 4.4 (The generalized vague associative law 1). Let M = (L,6,*)
stand for an integral, commutative cl-monoid. Given D 2 ORD~, if one of thefollowing two conditions is satisfied:
(a) ~ is transitive of the first and third order,Proof. (i) Taking the elements u1, . . .,un+m,v1, . . .,vm 2 X in Lemma 4.2(i) suchthat uk
Qki1aiRfDiji1;...;k1g and vj
Qmj1i1 anj1iLfiji1;...;nkg for all
k = 1, . . .,n + m and j = 1, . . .,m. Then, by Proposition 3.4, we have
WRa1; . . . ; anm; u1; . . . ; unm WRa1; . . . ; an; u1; . . . ; un WLan1; . . . ; anm; v1; . . . ; vm 1:
Therefore, by Lemma 4.2(i), we see that
1 WRa1; . . . ; anm; u1; . . . ; unm WLan1; . . . ; anm; v1; . . . ; vm6 WRa1; . . . ; an; u1; . . . ; un EunDv1; unm EunDv1; unm:
Thus we obtain
EYni1
ai
" #RfDi ji1;...;n1g
DYmi1
ani
" #Lfi ji1;...;m1g
;Ynmi1
ai
" #RfDi ji1;...;nm1g
0@ 1A EunDv1; unm 1;
so (i) is obvious.
Similar to (i), the equivalences (ii) and (iii) can be easily deduced from
Lemma 4.1(i) and (ii), respectively. h
As is noted in the preceding section, if the M-equivalence relation E is taken
as an M-equality on X in Proposition 4.3, then all exact right and left productsof given elements a1, . . .,an 2 X will be nothing but the product
Qni1ai of
a1, . . .,an in (X,), where ord~. In this case, the equivalence relation Ebecomes the classical equality relation = on X, and so all properties in Prop-
osition 4.3 turn to the generalized associative law in the classical sense. Conse-
quently, Proposition 4.3 gives a non-trivial result for only the case that E is an
M-equivalence relation, but not an M-equality.
Following the preparatory results stated in this section, we are now able to
formulate the generalized associative law for vague products. The concernedformulation will be given in the following two main theorems:
M. Demirci / Information Sciences 176 (2006) 900936 921(b) E is regular w.r.t. D,
If X ; ~ is assumed to be a perfect vague semigroup satisfying the propertiesthat the ordinary description of ~ contains an associative crisp binary opera-tion and ~ is transitive of the second or the third order, then it is an immediateconclusion of Proposition 3.6(iii) that the upper scripts R and L in allproperties in both Theorems 4.4 and 4.5 can be completely disregarded, i.e.
all properties in Theorems 4.4 and 4.5 become the same thing. This simple fact
is put into the concluding result:
Corollary 4.6. Let M = (L,6,*) be an integral, commutative cl-monoid, and let(a) ~ is transitive of the first and second order,(b) E is regular w.r.t. D,
is satisfied, then for n;m 2 N and for all a1, a2, . . ., an+m, x, y, z 2 X, the follow-ing properties are valid:
(i) ePLa1; a2; . . . ; anh ix ePLan1; an2; . . . ; anmh iyePLa1; a2; . . . ; anmh iz 6 ExDy; z.(ii) ePRa1; a2; . . . ; anh ix ePLan1; an2; . . . ; anmh iyePLa1; a2; . . . ; anmh iz 6 ExDy; z.
Proof. In an analogous manner to Theorem 4.4, (i) and (ii) can be easily
proven by using Lemma 4.1(ii) and Lemma 4.2(ii), respectively. For this
reason, their proofs are left to the readers as an exercise. hthen for n, m 2 N and for all a1,a2, . . ., an+m, x, y, z 2 X, the next properties aretrue:
(i) ePRa1; a2; . . . ; anh ix ePRan1; an2; . . . ; anmh iyePRa1; a2; . . . ; anmh iz 6 ExDy; z.(ii) ePRa1; a2; . . . ; anh ix ePLan1; an2; . . . ; anmh iyePRa1; a2; . . . ; anmh iz 6 ExDy; z.
Theorem 4.5 (The generalized vague associative law 2). Let M = (L,6,*)denote an integral, commutative cl-monoid. For a given D 2 ORD~, if one of theconditions:
922 M. Demirci / Information Sciences 176 (2006) 900936X ; ~ be a perfect M-vague semigroup w.r.t. P and E provided that there existsan associative ordinary description D 2 ORD~ of ~ and ~ is transitive of the
tary tools of classical algebra, and numerous algebraic structures have beensuccessfully built owing to these tools. In a parallel way to classical algebra,
many-valued counterparts to these tools therein play a similar role for the
development of vague algebraic structures. As a conrmation of this impres-
sion, integral powers of elements in vague groups will be derived as a particular
case of products of elements in vague groups, and their representation proper-
ties will be obtained from the present results in the companion paper [15]. Fur-second or the third order. Then for n, m 2 N and for all a1,a2, . . ., an+m, x, y,z 2 X, the following relation holds:ePa1; a2; . . . ; anh ix ePan1; an2; . . . ; anmh iy
ePa1; a2; . . . ; anmh iz 6 ExDy; z:5. Conclusion
In this paper, the notion of product of a nite number of elements and the
generalized associative law have been handled in vague semigroups by means
of three interlaced mathematical concepts: exact product, successive product
and vague product, where the rst two concepts are the essential components
of the third one. Because of the fact that representation properties of vague
products play a crucial role for the formulation of the properties of vague
products as well as their applications, they have also been introduced in this
paper. As a practical implementation of these representation properties, for suit-ably chosen M-equivalence relations ER2 on R
2 and ER on R and an M-vaguemultiplication (addition) operation ~ (~) w.r.t. ER2 and ER, it is pointed out thatthe vague product (sum) of given real numbers a1,a2, . . .,an w.r.t. ~ (~) can besimply calculated as the fuzzy equivalence class of a1 a2 an w.r.t. E (or theextensional hull of a1 a2 an w.r.t. E in the terminology of [24,25]).
Due to the discrete nature of scales of measurement instruments, there
exists an inevitable uncertainty on the indistinguishability of any two possible
values of a measured quantity q. Indistinguishability operators (or more gener-ally, M-equivalence relations) are natural mathematical tools to model this
uncertainty. If we take into account the indistinguishability operators repre-
senting this uncertainty, then it is demonstrated in this paper that a suitably
chosen M-vague multiplication (addition) operation ~ (~) w.r.t. these indistin-guishability operators becomes a mathematical representation of the multipli-
cation (addition) operation on the set of all possible values of q assumed to
be R, and the product (sums) of any possible values or measured values
a1,a2, . . .,an of q become the vague product (sum) of a1,a2, . . .,an w.r.t. ~ (~).Finite products of elements and the generalized associative law are rudimen-
M. Demirci / Information Sciences 176 (2006) 900936 923thermore, application of the generalized vague associative law to integral
powers of elements in vague groups will be a subject of Part II.
Appendix A. Proof of Lemma 4.1
(i) Since the required inequality is clear for the case u15 a1 or v15 an+1, weassume that u1 = a1 and v1 = an+1 without loss of the generality. Let the condi-
tion (a) be fullled. To conrm the claimed inequality, we apply induction on
m. For m = 1, we have
WRa1; . . . ; an1; u1; . . . ; un1 WRan1; v1 WRa1; . . . ; an1; u1; . . . ; un1 6 ~un; an1; un1 ~un; v1; un16 EunDv1; un1;
so the assertion is true for m = 1. As an induction hypothesis, let us suppose
that
WRa1; . . . ; anm1; u1; . . . ; unm1WRan1; . . . ; anm1; v1; . . . ; vm1
6 EunDvm1; unm1:
Then owing to the fact that ~ is a perfect M-vague binary operation w.r.t.P and E, we have
WRa1; . . . ; anm1; u1; . . . ; unm1WRan1; . . . ; anm1; v1; . . . ; vm1
6 EunDvm1; unm1 ~un; vm1; unm1: A:1Considering Theorem 2.3 and the inequality (A.1), and applying (VAS), we
observe that
WRa1; . . . ; anm; u1; . . . ; unm WRan1; . . . ; anm; v1; . . . ; vm WRa1; . . . ; anm1; u1; . . . ; unm1 ~unm1; anm; unm WRan1; . . . ; anm1; v1; . . . ; vm1 ~vm1; anm; vm
WRa1; . . . ; anm1; u1; . . . ; unm1WRan1; . . . ; anm1; v1; . . . ; vm1 ~unm1; anm; unm ~vm1; anm; vm
6 ~un; vm1; unm1 ~unm1; anm; unm ~vm1; anm; vm ~vm1; anm; vm ~un; vm; unDvm ~un; vm1; unm1 ~unm1; anm; unm
6 EunDvm; unm;
924 M. Demirci / Information Sciences 176 (2006) 900936hence the required inequality follows.
Now let us verify the assertion under the assumption (b). Because of the reg-
ularity of E w.r.t. D, it follows from Proposition 2.4(ii) that (X, vag(D)) is a per-fect M-vague semigroup w.r.t. P and E. Thus we directly deduce from the
previous case that
WRvagDa1; . . . ; anm; u1; . . . ; unm WRvagDan1; . . . ; anm; v1; . . . ; vm6 EunDvm; unm:
Finally, noting the inequalities
WRa1; . . . ; anm; u1; . . . ; unm 6 WRvagDa1; . . . ; anm; u1; . . . ; unmand WRan1; . . . ; anm; v1; . . . ; vm 6 WRvagDan1; . . . ; anm; v1; . . . ; vm;
the required inequality is got again.
(ii) If un+m5 an+m or vn5 an, the assertion is trivial. Assume that un+m =an+m and vn = an. Let us rst prove the required inequality for the case (a).For n = 1, we may write
WLa1; . . . ; am1; u1; . . . ; um1 WLa1; v1 WLa1; . . . ; am1; u1; . . . ; um1 ~a1; u2; u1 WLa2; . . . ; am1; u2; . . . ; um16 WLa2; . . . ; am1; u2; . . . ; um1 Ea1Du2; u1 WLa2; . . . ; am1; u2; . . . ; um1 Ev1Du2; u1;
so the required inequality is true for n = 1. Let us assume that nP 2.Using (VAS), and by the fact that ~ is a perfect M-vague binary operation
w.r.t. P and E, we may write
~vk; un1; uk ~ak1; uk; uk1 ~ak1; vk; vk1 ~vk; un1; uk ~ak1; uk; uk1 ~ak1; vk; vk1 ~vk1; un1; vk1Dun1
6 Euk1; vk1Dun1 ~vk1; un1; uk1 A:2for all k = 2, . . .,n. For nP 3 and for all k = 3, . . .,n, exploiting (A.2), weobserve that
~vk; un1; uk ~a1; u2; u1 ~a2; u3; u2 ~ak1; uk; uk1 ~a1; v2; v1 ~a2; v3; v2 ~ak1; vk; vk1
~vk; un1; uk ~ak1; uk; uk1 ~ak1; vk; vk1 ~a1; u2; u1 ~a2; u3; u2 ~ak2; uk1; uk2
M. Demirci / Information Sciences 176 (2006) 900936 925 ~a1; v2; v1 ~a2; v3; v2 ~ak2; vk1; vk2
6 ~vk1; un1; uk1 ~a1; u2; u1 ~a2; u3; u2 ~ak2; uk1; uk2 ~a1; v2; v1 ~a2; v3; v2 ~ak2; vk1; vk2: A:3
Then for nP 3, recalling the fact vn = an, and applying the inequalities (A.2)and (A.3), we estimate the following inequalities
~a1; u2; u1 ~a2; u3; u2 ~an; un1; un WLa1; . . . ; an; v1; . . . ; vn ~vn; un1; un ~a1; u2; u1 ~a2; u3; u2 ~an1; un; un1 ~a1; v2; v1 ~a2; v3; v2 ~an1; vn; vn1
6 ~vn1; un1; un1 ~a1; u2; u1 ~a2; u3; u2 ~an2; un1; un2 ~a1; v2; v1 ~a2; v3; v2 ~an2; vn1; vn2 6 ~vn2; un1; un2 ~a1; u2; u1 ~a2; u3; u2 ~an3; un2; un3 ~a1; v2; v1 ~a2; v3; v2 ~an3; vn2; vn3
6 6 ~v2; un1; u2 ~a1; u2; u1 ~a1; v2; v1 6 Eu1; v1Dun1:Hence we get
~a1; u2; u1 ~a2; u3; u2 ~an; un1; unWLa1; . . . ; an; v1; . . . ; vn
6 Eu1; v1Dun1: A:4
for nP 3. If n = 2, the inequality (A.4) will be nothing but the inequality (A.2)for k = 2, so the inequality (A.4) is true for nP 2.
On the other hand, we easily deduce from the denition of the map WL theequality
WLa1; . . . ;anm; u1; . . . ;unm ~a1;u2;u1 ~a2;u3;u2 ~an;un1;unWLan1; . . . ;anm; un1; . . . ;unm:
At last, if we multiply both sides of the inequality (A.4) by WLan1; . . . ; anm,(un+1, . . .,un+m)] w.r.t. *, we see that
WLa1; . . . ; anm; u1; . . . ; unm WLa1; . . . ; an; v1; . . . ; vn WLan1; . . . ; anm; un1; . . . ; unm ~a1; u2; u1 ~a2; u3; u2 ~an; un1; un WLa1; . . . ; an; v1; . . . ; vn
926 M. Demirci / Information Sciences 176 (2006) 9009366 WLan1; . . . ; anm; un1; . . . ; unm Eu1; v1Dun1:
1 n1 1 n1 WRa1; . . . ; an; u1; . . . ; un ~un; an1; un16 WRa1; . . . ; an; u1; . . . ; un EunDan1; un1 WRa1; . . . ; an; u1; . . . ; un EunDv1; un1;
so the assertion is obvious. Consider the case mP 2 and n 2 N. Applying thevague associativity (VAS), we may write
~unk; ank1; unk1 ~ank1; vk2; vk1 ~ank1; vk2; vk1 ~unk; vk1; unkDvk1 ~unk; ank1; unk1 ~unk1; vk2; unk1Dvk2Appendix B. Proof of Lemma 4.2
(i) The assertion is trivial for the case u15 a1 or vm5 an+m. Suppose thatu1 = a1 and vm = an+m. If m = 1 and n 2 N, we may write
WRa1; . . . ; an1; u1; . . . ; un1 WLan1; v1 WRa ; . . . ; a ; u ; . . . ; u Now we prove the assertion under the assumption (b). Before the establishment
of the proof for the case (b), it will be useful to notice that the steps after the
inequality (A.4) in the proof for the case (a) are independent from the assump-
tions both (a) and (b). For this reason, it is sucient to conrm the inequality
(A.4) under the assumption (b). Then the required inequality is easily obtained
by simply repeating the same arguments just after (A.4) in the previous case. Inorder to verify (A.4) under the assumption (b), we employ the same technique
as in the property (i). From the denition of the maps WL, WLvagD, we obvi-ously have WL 6 WLvagD. By the fact that E is regular w.r.t. D, it is clear fromProposition 2.4(ii) that (X, vag(D)) is a perfectM-vague semigroup w.r.t. P andE. Therefore, applying the previous case, we easily see that
~a1; u2; u1 ~a2; u3; u2 ~an; un1; unWLa1; . . . ; an; v1; . . . ; vn6 vagDa1; u2; u1 vagDa2; u3; u2 vagDan; un1; unWLvagDa1; . . . ; an; v1; . . . ; vn
6 Eu1; v1Dun1:
M. Demirci / Information Sciences 176 (2006) 900936 9276 EunkDvk1; unk1Dvk2 B:1
for all k = 0,1, . . .,m 2. Since the inequality (B.1) is valid for allk = 0,1, . . .,m 2, the inequality (B.1) implies the existence of the followingset of inequalities:
~un;an1;un1 ~an1;v2;v16EunDv1;un1Dv2;~un1;an2;un2 ~an2;v3;v26Eun1Dv2;un2Dv3;...
~unm2;anm1;unm1 ~anm1;vm;vm16Eunm2Dvm1;unm1Dvm:
Now multiplying all of these inequalities side by side w.r.t. *, we obtain that
~un; an1; un1 ~un1; an2; un2 ~unm2; anm1; unm1 ~an1; v2; v1 ~an2; v3; v2 ~anm1; vm; vm1
6 EunDv1; un1Dv2 Eun1Dv2; un2Dv3 Eunm2Dvm1; unm1Dvm 6 EunDv1; unm1Dvm:
Therefore, considering the fact
~un; an1; un1 ~un1; an2; un2 ~unm2; anm1; unm1 ~an1; v2; v1 ~an2; v3; v2 ~anm1; vm; vm1
~un; an1; un1 ~un1; an2; un2 ~unm2; anm1; unm1WLan1; . . . ; anm; v1; . . . ; vm;
we reach the inequality
~un; an1; un1 ~un1; an2; un2 ~unm2; anm1; unm1WLan1; . . . ; anm; v1; . . . ; vm
6 EunDv1; unm1Dvm: B:2On the other hand, since vm = an+m, we obviously have
~unm1; anm; unm ~unm1; vm; unm 6 Eunm1Dvm; unm: B:3The inequalities (B.2) and (B.3) entail that
~un; an1; un1 ~un1; an2; un2 ~unm1; anm; unmWLan1; . . . ; anm; v1; . . . ; vm
6 EunDv1; unm1Dvm Eunm1Dvm; unm 6 EunDv1; unm: B:4Finally, using the fact
WRa1; . . . ; anm; u1; . . . ; unm WRa1; . . . ; an; u1; . . . ; un ~un; an1; un1 ~un1; an2; un2
928 M. Demirci / Information Sciences 176 (2006) 900936 ~unm1; anm; unm;
and multiplying both sides of the inequality (B.4) by WRa1; . . . ; an;u1; . . . ; un w.r.t. *, we see that
WRa1; . . . ; anm; u1; . . . ; unm WLan1; . . . ; anm; v1; . . . ; vm WRa1; . . . ; an; u1; . . . ; un ~un; an1; un1 ~un1; an2; un2 ~unm1; anm; unm WLan1; . . . ; anm; v1; . . . ; vm
6 WRa1; . . . ; an; u1; . . . ; un EunDv1; unm;and hence the assertion is now straightforward.
(ii) Similar to (i), we assume that u1 = a1 and vn+m = an+m without loss of the
generality. For n = 1, the validity of the claimed inequality can be seen in an
analogous manner to (i), so it is omitted here. Suppose that nP 2. Invokingthe inequality (B.1), we can establish the following set of inequalities:
~un1; an; un ~an; vn1; vn 6 Eun1Dvn; unDvn1;~un2; an1; un1 ~an1; vn; vn1 6 Eun2Dvn1; un1Dvn;...
~u1; a2; u2 ~a2; v3; v2 6 Eu1Dv2; u2Dv3:
In a similar fashion to (i), multiplying all of these inequalities side by side w.r.t.
*, we observe that
WRa1; . . . ; an; u1; . . . ; un ~a2; v3; v2 ~an1; vn; vn1 ~an; vn1; vn
~un1; an; un ~an; vn1; vn ~un2; an1; un1 ~an1; vn; vn1 ~u1; a2; u2 ~a2; v3; v26 Eun1Dvn; unDvn1 Eun2Dvn1; un1Dvn Eu1Dv2; u2Dv36 EunDvn1; u1Dv2:
Then using this inequality, and considering the fact
~a1; v2; v1 ~u1; v2; v1 6 Eu1Dv2; v1;we see that
WRa1; . . . ; an; u1; . . . ; un ~a1; v2; v1 ~a2; v3; v2 ~an1; vn; vn1 ~an; vn1; vn6 EunDvn1; u1Dv2 Eu1Dv2; v1 6 EunDvn1; v1; B:5
L
M. Demirci / Information Sciences 176 (2006) 900936 929and multiplying both sides of the inequality (B.5) by W an1; . . . ; anm;vn1; . . . ; vnm w.r.t. *, we obtain that
WR an1; . . . ; anm; an1;Y2
ani ;Y3
ani ; . . . ;@4
930 M. Demirci / Information Sciences 176 (2006) 900936i1 fD1g i1 fDi ji1;2gYm1i1
ani
" #RfDi ji1;...;m2g
; y
1A 6 E Yni1
ai
" #Rfi ji1;...;n1g
Dy; z
0@ 1A35:Thus, by using the fact that ~ is a perfect M-vague binary operation w.r.t. Pand E, we see that
~Ynm1i1
ai
" #Rfi ji1;...;nm2g
; anm; z
0@ 1A ~ Ym1i1
ani
" #RfDiji1;...;m2g
; anm; y
0@ 1A6 E
Ynai
" #RDy; z
0@ 1A ~ Yn ai" #R
; y; z
0@ 1A:WLa1; . . . ; anm; v1; . . . ; vnm WRa1; . . . ; an; u1; . . . ; un ~a1; v2; v1 ~a2; v3; v2 ~an1; vn; vn1 ~an; vn1; vnWLan1; . . . ; anm; vn1; . . . ; vnmWRa1; . . . ; an; u1; . . . ; un
WLan1; . . . ; anm; vn1; . . . ; vnm WRa1; . . . ; an; u1; . . . ; un ~a1; v2; v1 ~a2; v3; v2 ~an1; vn; vn1 ~an; vn1; vn
6 WLan1; . . . ; anm; vn1; . . . ; vnm EunDvn1; v1;
and hence the proof is now completed. h
Appendix C. Proof of Theorem 4.4
(i) To prove the required inequality, we invoke Lemma 4.1(i). Let us rst as-sume that (a) is fullled. For {iji = 1, . . .,n + m2}, fDiji 1; . . . ;m 2g ORD~, putting un+m = z, vm = y, uk
Qki1aiRfi ji1;...;k1g and vl
Qli1aniRfDiji1;...;l1g for all k = 1, . . .,n + m 1 and l = 1, . . .,m 1 in Lemma4.1(i), and utilizing the equality (3.1), we obtain from Lemma 4.1(i) that
~Ynm1i1
ai
" #Rfi ji1;...;nm2g
; anm; z
0@ 1A ~ Ym1i1
ani
" #RfDi ji1;...;m2g
; anm; y
0@ 1A WR a1; . . . ; anm; a1;
Y2i1
ai
" #Rf1g
;Y3i1
ai
" #Rfi ji1;2g
; . . . ;Ynm1i1
ai
" #Rfi ji1;...;nm2g
; z
0@ 1A24 35" #R " #R02i1 fi ji1;...;n1g i1 fiji1;...;n1g
Then, for fUiji 1; . . . ; n 1g ORD~, multiplying both sides of thisinequality by ~Qn1i1 aiRfUi ji1;...;n2g; an; x w.r.t. *, and by making use of the rstorder transitivity of ~, we observe that
~Yn1i1
ai
" #RfUi ji1;...;n2g
; an; x
0@ 1A ~ Ym1i1
ani
" #RfDi ji1;...;m2g
; anm; y
0@ 1A
~Ynm1i1
ai
" #Rfiji1;...;nm2g
; anm; z
0@ 1A
6 ~Yn1i1
ai
" #RfUi ji1;...;n2g
; an; x
0@ 1A ~ Yni1
ai
" #Rfiji1;...;n1g
; y; z
0@ 1A
EYn1i1
ai
" #RfUi ji1;...;n2g
Un1an; x
0@ 1A ~ Yni1
ai
" #Rfiji1;...;n1g
; y; z
0@ 1A
EYni1
ai
" #RfUi ji1;...;n1g
; x
0@ 1A ~ Yni1
ai
" #Rfiji1;...;n1g
; y; z
0@ 1A: C:1On the other hand, we have from Proposition 3.2 that
EYni1
ai
" #RfUi ji1;...;n1g
;Yni1
ai
" #Rfi ji1;...;n1g
0@ 1A 1:Then, we may write
EYni1
ai
" #RfUi ji1;...;n1g
;x
0@ 1AE Yni1
ai
" #RfUi ji1;...;n1g
;x
0@ 1AE
Yni1
ai
" #RfUiji1;...;n1g
;Yni1
ai
" #Rfiji1;...;n1g
0@ 1A6E
Yni1
ai
" #Rfi ji1;...;n1g
;x
0@ 1A;
M. Demirci / Information Sciences 176 (2006) 900936 931and reciprocally,
EYni1
ai
" #Rfiji1;...;n1g
; x
0@ 1A 6 E Yni1
ai
" #RfUi ji1;...;n1g
; x
0@ 1A;so the equality
EYni1
ai
" #RfUi ji1;...;n1g
; x
0@ 1A E Yni1
ai
" #Rfi ji1;...;n1g
; x
0@ 1Afollows. Therefore, substituting EQni1aiRfUiji1;...;n1g; x byEQni1aiRfiji1;...;n1g; x in the inequality (C.1), we may write
~Yn1i1
ai
" #RfUi ji1;...;n2g
; an; x
0@ 1A ~ Ym1i1
ani
" #RfDi ji1;...;m2g
; anm; y
0@ 1A ~
Ynm1i1
ai
" #Rfi ji1;...;nm2g
; anm; z
0@ 1A6 E
Yni1
ai
" #Rfi ji1;...;n1g
; x
0@ 1A ~ Yni1
ai
" #Rfiji1;...;n1g
; y; z
0@ 1A6 ~x; y; z 6 ExDy; z: C:2
Since * is distributive over arbitrary joins, and by the inequality (C.2), we ob-
serve that
ePRa1; a2; . . . ; anh ix ePRan1; an2; . . . ; anmh iy ePRa1; a2; . . . ; anmh iz
_
~Yn1i1
ai
" #RfUi ji1;...;n2g
; an; x
0@ 1AfUiji 1; . . . ; n 2g ORD~8
~Ynm1i1
ai
" #Rfi ji1;...;nm2g
; anm; z
0@ 1AjfUiji 1; . . . ; n 2g;fDiji 1; . . . ;m 2g; fiji 1; . . . ; n m 2g ORD~
9=;6 ExDy; z:
In order to complete the proof, we nally assume that the condition (b) is true. For
the verication of the assertion, we use the same technique as in Lemma 4.1(i).
Owing to Proposition 2.4(i) and (ii), (X,vag()) is an M-vague semigroup, andvag() is transitive of the rst and third order. Thus, the previous case gives thatePRa1; a2; . . . ; anh ix ePRan1; an2; . . . ; anmh iy
ePRa1; a2; . . . ; anmh iz6 ePRvaga1; a2; . . . ; anh ix ePRvagan1; an2; . . . ; anmh iy ePRvaga1; a2; . . . ; anmh iz
6 ExDy; z:(ii) Let (a) be satised. For {iji = 1, . . .,n + m 2}, fDiji 1; . . . ;m 2g
ORD~, choosing the elements u1, . . .,un+m,v1, . . ., vm 2 X in Lemma 4.2(i)such that un+m = z, v1 = y, uk
Qki1aiRfiji1;...;k1g and vl
Qml1i1 anl1iLfDi ji1;...;mlg for all k = 1, . . .,n + m 1 and l = 2, . . .,m, andusing the equalities (3.1) and (3.2), we may write
~Ynm1i1
ai
" #Rfi ji1;...;nm2g
;anm;z
0@ 1A ~ an1; Ym1i1
an1i
" #LfDi ji1;...;m2g
;y
0@ 1AWR a1; . . . ;anm; a1;
Y2i1
ai
" #Rf1g
;Y3i1
ai
" #Rfi ji1;2g
; . . . ;Ynm1i1
ai
" #Rfi ji1;...;nm2g
;z
0@ 1A24 35WL an1; . . . ;anm; y;
Ym1i1
an1i
" #LfDi ji1;...;m2g
;Ym2i1
an2i
" #LfDi ji1;...;m3g
; . . . ;
0@24Y2i1
anm2i
" #LfD1g
;anm
1A356WRa1; . . . ;an; a1;
Y2i1
ai
" #Rf1g
; . . . ;Yni1
ai
" #Rfi ji1;...;n1g
0@ 1A24 35E
Yni1
ai
" #Rfi ji1;...;n1g
Dy;z
0@ 1A 1E Yni1
ai
" #Rfi ji1;...;n1g
Dy;z
0@ 1AYn" #R0 1
M. Demirci / Information Sciences 176 (2006) 900936 933Ei1
aifi ji1;...;n1g
Dy;z@ A:
Ynm1" #R Ym1" #L
6 ~ ai ; an; x@ A ~ ai ; y; z@ A
i1 fUi ji1;...;n1g i1 fiji1;...;n1g
Similar to (i), taking supremum over {Uiji = 1, . . .,n 2}, {Diji = 1, . . .,m 2}
and {iji = 1, . . .,n + m 2} in the both sides of the inequality (C.3), andexploiting the distributivity of * over arbitrary joins, the required inequality EYni1
ai
" #Rfi ji1;...;n1g
; x
0@ 1A ~ Yni1
ai
" #Rfiji1;...;n1g
; y; z
0@ 1A6 ~x; y; z 6 ExDy; z: C:3i1 fUi ji1;...;n2g i1 fiji1;...;n1g
EYn
ai
" #R; x
0@ 1A ~ Yn ai" #R
; y; z
0@ 1A~i1
aifi ji1;...;nm2g
; anm; z@ A ~ an1;i1
an1ifDiji1;...;m2g
; y@ A
6 EYni1
ai
" #Rfiji1;...;n1g
Dy; z
0@ 1A ~ Yni1
ai
" #Rfi ji1;...;n1g
; y; z
0@ 1A:In a similar fashion to (i), for fUiji 1; . . . ; n 1g ORD~, multiplyingboth sides of this inequality by ~Qn1i1 aiRfUiji1;...;n2g; an; x w.r.t. *, and apply-ing the rst and third order transitivity of ~, we easily see that
~Yn1i1
ai
" #RfUi ji1;...;n2g
; an; x
0@ 1A ~ an1; Ym1i1
an1i
" #LfDiji1;...;m2g
; y
0@ 1A
~Ynm1i1
ai
" #Rfi ji1;...;nm2g
; anm; z
0@ 1AYn1" #R0 1 Yn" #R0 1Thus, by using the fact that ~ is a perfect M-vague binary operation w.r.t.P and E, we get
0 1 0 1
934 M. Demirci / Information Sciences 176 (2006) 900936is easily acquired from the inequality (C.3). The proof of the required inequal-
ity under the assumption (b) is analogous to the property (i). h
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The generalized associative law in vague groups and its applications mdash IIntroductionA brief introduction to vague semigroupsProducts of elements in vague groups and their fundamental propertiesThe generalized associative law in vague semigroupsConclusionProof of Lemma 4.1Proof of Lemma 4.2Proof of Theorem 4.4References