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The generalized associative law in vague groups and its applications—I 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- 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 generalized 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 applications—II, 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 finite 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 finite number of real numbers in vague arithmetic, some practical applications of the notions of product and sum of a finite number of real numbers in vague arithme- tic are also given. Ó 2005 Elsevier Inc. All rights reserved. 0020-0255/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2005.01.020 * Fax: +90 242 227 8911. E-mail address: [email protected] Information Sciences 176 (2006) 900–936 www.elsevier.com/locate/ins

The generalized associative law in vague groups and its applications—I

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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

  • References

    [1] R. Belohlavek, Similarity relations and BK-relational products, Information Sciences 126

    (2000) 287295.

    [2] U. Bodenhofer, Similarity-based generalization of fuzzy orderings preserving the classical

    axioms, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 8 (3)

    (2000) 593610.

    [3] D. Boixader, J. Jacas, J. Recasens, Fuzzy equivalence relations: advanced material, in: D.

    Dubois, H. Prade (Eds.), Fundamentals of Fuzzy Sets, The Handbooks of Fuzzy Sets Series,

    vol. 7, Kluwer Academic Publishers, Boston, 2000, pp. 261290.

    [4] B. De Baets, R. Mesiar, Pseudo-metrics andT-equivalences, Journal of Fuzzy Mathematics 5

    (1997) 471481.

    [5] B. De Baets, R. Mesiar, Metrics and T-equalities, Journal of Mathematical Analysis and

    Applications 267 (2002) 531547.

    [6] B. De Baets, M. Mares, R. Mesiar, T-partitions of the real line generated by idempotent

    shapes, Fuzzy Sets and Systems 91 (1997) 177184.

    [7] M. Demirci, Fuzzy functions and their applications, Journal of Mathematical Analysis and

    Applications 252 (2000) 495517.

    [8] M. Demirci, Fundamentals of M-vague algebra and M-vague arithmetic operations,

    International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 10 (1)

    (2002) 2575.

    [9] M. Demirci, Foundations of fuzzy functions and vague algebra based on many-valued

    equivalence relations, Part I: Fuzzy functions and their applications, International Journal of

    General Systems 32 (2) (2003) 123155.

    [10] M. Demirci, Foundations of fuzzy functions and vague algebra based on many-valued

    equivalence relations, Part II: Vague algebraic notions, International Journal of General

    Systems 32 (2) (2003) 157175.

    [11] M. Demirci, Foundations of fuzzy functions and vague algebra based on many-valued

    equivalence relations, Part III: Constructions of vague algebraic notions and vague arithmetic

    operations, International Journal of General Systems 32 (2) (2003) 177201.

    [12] M. Demirci, Arithmetic of fuzzy quantities based on vague arithmetic operations, fuzzy sets

    and systems-IFSA 2003, Proceedings Lecture Notes in Articial Intelligence 2715 (2003) 159

    166.

    [13] M. Demirci, Indistinguishability operators in measurement theory, Part I: Conversions of

    indistinguishability operators with respect to scales, International Journal of General Systems

    32 (5) (2003) 415430.

    [14] M. Demirci, The generalized associative law in smooth groups, Information Sciences 169

    (2005) 227244.

    [15] M. Demirci, The generalized associative law in vague groups and its applicationsII,

    Information Sciences (submitted for publication).

    [16] M. Demirci, Products of elements in vague semigroups and their implementations in vague

    arithmetic, Fuzzy Sets and Systems (submitted for publication).

    [17] U. Hohle, N. Blanchard, Partial ordering in L-underdeterminate sets, Information Sciences 35

    (1985) 133144.

    [18] U. Hohle, Quotients with respect to similarity relations, Fuzzy Sets and Systems 27 (1988) 31

    44.

    [19] U. Hohle, Commutative, residuated l-monoids, in: U. Hohle, E.P. Klement (Eds.), Non-

    classical Logics and Their Applications to Fuzzy Subsets, Kluwer Academic Publishers,

    Boston, Dordrecht, 1995, pp. 53106.

    M. Demirci / Information Sciences 176 (2006) 900936 935[20] U. Hohle, Many-valued equalities, singletons and fuzzy partitions, Soft Computing 2 (1998)

    134140.

  • [21] U. Hohle, Classication of subsheaves over GL-algebra, in: Logic Colloquium98, LectureNotes in Logic 13, Springer-Verlag, 1998.

    [22] U. Hohle, A.P. Sostak, Axiomatic foundations of xed-basis fuzzy topology, in: U. Hohle,

    S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory,

    The Hand Books of Fuzzy Sets Series, vol. 3, Kluwer Academic Publishers, Boston,

    Dordrecht, 1999, pp. 123273.

    [23] J. Jacas, J. Recasens, Fuzzy numbers and equality relations, in: Proceedings of FUZZIEEE93, San Francisco, 1993, pp. 12981301.

    [24] F. Klawonn, J.L. Castro, Similarity in fuzzy reasoning, Mathware and Soft Computing 2

    (1995) 1972281.

    [25] F. Klawonn, Fuzzy points, fuzzy relations and fuzzy functions, in: V. Novak, I. Perlieva

    (Eds.), Discovering World with Fuzzy Logic, Physica-Verlag, Heidelberg, 2000, pp. 431453.

    [26] D.H. Krantz, R.D. Luce, P. Suppes, A. Tversky, Foundations of Measurement, vol. 1,

    Academic Press, San Diego, 1971.

    [27] J.N. Mordeson, D.S. Malik, Fuzzy Commutative Algebra, World Scientic Publishing Co.

    Pte. Ltd, Singapore, 1998.

    [28] S. Sezer, Vague groups and generalized vague subgroups on the basis of ([0,1],6,^),Information Sciences (in press).

    [29] L. Valverde, On the structure of F-indistinguishability operators, Fuzzy Sets and Systems 17

    (1985) 313328.

    [30] L.A. Zadeh, Similarity relations and fuzzy orderings, Information Sciences 3 (1971) 177200.

    936 M. Demirci / Information Sciences 176 (2006) 900936

    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