43
The generalized associative law in vague groups and its applications—II Mustafa Demirci * Akdeniz University, Faculty of Sciences and Arts, Department of Mathematics, 07058-Antalya, Turkey Received 25 May 2004; received in revised form 22 February 2005; accepted 26 February 2005 Abstract As a continuation of Part I, vague integral powers of elements in vague groups and their representation properties are introduced in this paper. Thereafter, some rudimen- tary algebraic properties of vague integral powers of elements, obtained from the gen- eralized vague associative law formulated in Part I, are established.The present paper particularly provides the abstract foundations of integral powers and multiples of real numbers in vague arithmetic. For this reason, special attention is also paid to the calcu- lation of integral powers and multiples of real numbers in vague arithmetic, and some practical applications related to the discrete structure of measurement instruments are also given. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Fuzzy groups; Vague groups; Fuzzy arithmetic; Vague arithmetic; Fuzzy equivalence relations; Fuzzy equalities 0020-0255/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2005.02.007 * Fax: +90 242 2278911. E-mail address: [email protected] Information Sciences 176 (2006) 1488–1530 www.elsevier.com/locate/ins

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

Embed Size (px)

Citation preview

Information Sciences 176 (2006) 1488–1530

www.elsevier.com/locate/ins

The generalized associative law invague groups and its applications—II

Mustafa Demirci *

Akdeniz University, Faculty of Sciences and Arts, Department of Mathematics,

07058-Antalya, Turkey

Received 25 May 2004; received in revised form 22 February 2005; accepted 26 February 2005

Abstract

As a continuation of Part I, vague integral powers of elements in vague groups and

their representation properties are introduced in this paper. Thereafter, some rudimen-

tary algebraic properties of vague integral powers of elements, obtained from the gen-

eralized vague associative law formulated in Part I, are established.The present paper

particularly provides the abstract foundations of integral powers and multiples of real

numbers in vague arithmetic. For this reason, special attention is also paid to the calcu-

lation of integral powers and multiples of real numbers in vague arithmetic, and some

practical applications related to the discrete structure of measurement instruments are

also given.

� 2005 Elsevier Inc. All rights reserved.

Keywords: Fuzzy groups; Vague groups; Fuzzy arithmetic; Vague arithmetic; Fuzzy equivalence

relations; Fuzzy equalities

0020-0255/$ - see front matter � 2005 Elsevier Inc. All rights reserved.

doi:10.1016/j.ins.2005.02.007

* Fax: +90 242 2278911.

E-mail address: [email protected]

M. Demirci / Information Sciences 176 (2006) 1488–1530 1489

1. Introduction

The necessity of a many-valued counterpart to the notion of integral power

of elements in vague groups for developing vague algebraic structures is ex-

pressed in Part I. For this purpose, as a continuation of Part I, the present

study introduces integral powers of elements in vague groups, and focuseson their elementary properties. Integral powers of elements in vague groups

will be considered in three interlaced mathematical notions: exact integral

power, successive integral power and vague integral power of elements, where

the third notion represents the required many-valued counterpart to integral

power of elements in vague groups and constitutes the first two notions. Alge-

braic properties, derived from the generalized vague associative law in Part I,

and representation properties of vague integral powers of elements will be main

subjects of this paper. The content of this paper can be briefly described as fol-lows: after this introductory section, a definition of the vague integral power of

elements in vague groups and its subsidiary tools will be given in Section 2.

Furthermore, representation properties of vague integral powers of elements

and their application to the calculation of vague integral powers and multiples

of real numbers in vague arithmetic will be introduced in Section 2. Realiza-

tions of vague integral powers and multiples of real numbers in practical mea-

surements will also be the subject of Section 2. In Section 3, some fundamental

algebraic properties of vague integral powers of elements will be establishedusing the generalized vague associative law in Part I.

2. Integral powers of elements in vague groups and their fundamental properties

We first start with some necessary preliminaries pertained to vague groups,

and we refer to the companion paper [3] for other necessary preliminary

notions. As in [3], the triple M = (L,6, *) always stands for an integral, com-mutative cqm-lattice unless otherwise mentioned in this paper. Furthermore,

X always denotes a nonempty set.

Definition 2.1 [1]. Let P and E be M-equivalence, relations on X · X and X,

respectively. Given an M-vague binary operation ~� on X w.r.t. P and E, the

ordered pair ðX ; ~�Þ is called an M-vague semigroup w.r.t. P and E iff ~� satisfies

the condition of vague associativity:

(VAS) ð8a; b; c; d;m; q;w 2 X Þð~�ðb; c; dÞ � ~�ða; d;mÞ � ~�ða; b; qÞ � ~�ðq; c;wÞ6 Eðm;wÞÞ.

An M-vague semigroup ðX ; ~�Þ w.r.t. P and E is called an M-vague monoid

w.r.t. P and E iff it satisfies the condition:

1490 M. Demirci / Information Sciences 176 (2006) 1488–1530

(VID) There exists e 2 X (called an identity element of ðX ; ~�Þ) such that~�ðe; a; aÞ ¼ ~�ða; e; aÞ ¼ 1 for all a 2 X.

An M-vague monoid ðX ; ~�Þ w.r.t. P and E is called an M-vague group w.r.t. P

and E iff the following condition is fulfilled:

(VIN) There exists at least one identity element e of ðX ; ~�Þ such that for all

a 2 X, there exists an inverse element a�1 of a w.r.t. e, i.e. ~�ða�1; a; eÞ ¼~�ða; a�1; eÞ ¼ 1.

An M-vague group ðX ; ~�Þ w.r.t. P and E is said to be perfect iff ~� is a perfect

M-vague binary operation on X w.r.t. P and E. An identity element in a vague

monoid ðX ; ~�Þ is not unique in general, and the set of all identity elements of

ðX ; ~�Þ is denoted by IdðX ; ~�Þ. If an identity element e of the vague monoid

ðX ; ~�Þ satisfies the condition (VIN), it is called an identity element of the vague

group ðX ; ~�Þ. We will denote the set of all identity elements of a vague group

ðX ; ~�Þ by Id�ðX ; ~�Þ. Similarly, if we consider a vague group ðX ; ~�Þ, an inverse

of a 2 X w.r.t. e 2 Id�ðX ; ~�Þ is not unique in general, and the set of all inversesof a w.r.t. e is represented by InvðX ;~�Þða; eÞ. Vague groups have numerous inter-

esting properties [1,2]. Some of these properties, which will be needed in the

sequel, are gathered in the following proposition.

Proposition 2.2. Let ðX ; ~�Þ be an M-vague group w.r.t. P and E. Then for all

a,x,y,z 2 X,e, e0 2 Id�ðX ; ~�Þ; b 2 InvðX ;~�Þða; eÞ; b0 2 InvðX ;~�Þða; e0Þ; d 2 InvðX ;~�Þðb; eÞ and � 2 ORDð~�Þ, the following statements are true:

(i) x � (y � z) �E (x � y) � z [4].

(ii) a �E d.

(iii) If ~� is a perfect M-vague binary operation, or if there exists at least one

D 2 ORDð~�Þ such that E is regular w.r.t. D, then b �E b 0.

(iv) vag(�) is transitive of the third (second) order iff E is right (left) regular

w.r.t. � [3].

(v) If E is regular w.r.t. �, then (X,vag(�)) is a perfect M-vague group w.r.t P

and E [3].(vi) If ðX ; ~�Þ is a perfect M-vague group w.r.t. P and E, then ~� is completely

transitive.

(vii) If (X, vag(�)) is an M-vague group w.r.t. P and E, then Id�ðX ; ~�Þ �Id�ðX ; vagð�ÞÞ and InvðX ;~�Þða; eÞ � InvðX ;vagð�ÞÞða; eÞ.

(viii) If ~� is a perfect M-vague binary operation, or if E is regular w.r.t. D for

some D 2 ORDð~�Þ, then E(x,y) = E(u, v) for all u 2 InvðX ;~�Þðx; eÞ and

v 2 Invðx;~�Þðy; eÞ.

M. Demirci / Information Sciences 176 (2006) 1488–1530 1491

Proof. (ii) Using the condition (VAS) of vague associativity, we have

1 ¼ ~�ðb; d; eÞ � ~�ða; e; aÞ � ~�ða; b; eÞ � ~�ðe; d; dÞ 6 Eða; dÞ; i:e: a�Ed:

(iii) If ~� is a perfect M-vague binary operation, the assertion is animmediate consequence of [1, Proposition 2.9(iv)]. Let us consider

the case: E is regular w.r.t. D for some D 2 ORDð~�Þ. Since

b 2 InvðX ;~�Þða; eÞ, and using the regularity of E w.r.t. D, we observe

that

~�ðb; a; eÞ ¼ 1 6 EðbDa; eÞ 6 EððbDaÞDb0; eDb0Þand

~�ðe; b0; b0Þ ¼ 1 6 EðeDb0; b0Þ) 1 6 EððbDaÞDb0; eDb0Þ � EðeDb0; b0Þ 6 EððbDaÞDb0; b0Þ) ðbDaÞDb0�Eb0:

Similarly, considering the fact b0 2 InvðX ;~�Þða; e0Þ, and by making use

of the regularity of E w.r.t. D, we easily get bD(aDb 0) �E b. On the

other hand, from (i), we have the equivalence bD(aD b 0) �E

(bDa)Db 0. Thus b �E b 0 follows from the equivalences bD(aDb 0)

�E (bDa)Db 0, (bDa)Db 0 �E b 0 and bD(aDb 0) �E b, at once.

(vi) Let ðX ; ~�Þ be a perfect M-vague group w.r.t. P and E. To show the

complete transitivity of ~�, it is sufficient to see the second and thirdorder transitivity of ~�. Let us pick x,y,z,u 2 X. Since ðX ; ~�Þ is an

M-vague group, it is clear that 9e 2 Id�ðX ; ~�Þ; 9x�1 2 InvðX ;~�Þðx; eÞand 9y�1 2 InvðX ;~�Þðy; eÞ. Then using the condition (VAS), and by

the fact that ~� is a perfect M-vague binary operation w.r.t. P and E,

we can write

~�ðx; y; zÞ ¼ ~�ðx; y; zÞ � ~�ðx�1; z; x�1 � zÞ � ~�ðx�1; x; eÞ � ~�ðe; y; yÞ6 Eðx�1 � z; yÞ ¼ ~�ðx�1; z; yÞ

and

~�ðx�1; z; yÞ ¼ ~�ðx�1; z; yÞ � ~�ðx; y; x � yÞ � ~�ðx; x�1; eÞ � ~�ðe; z; zÞ6 Eðx � y; zÞ ¼ ~�ðx; y; zÞ:

Thus we get

~�ðx; y; zÞ ¼ ~�ðx�1; z; yÞ: ð2:1ÞThen since ~� is a perfect M-vague binary operation w.r.t. P and E,

and applying (2.1), we observe that

~�ðx; y; zÞ � Eðy; uÞ ¼ ~�ðx�1; z; yÞ � Eðy; uÞ 6 ~�ðx�1; z; uÞ:

1492 M. Demirci / Information Sciences 176 (2006) 1488–1530

Furthermore due to the fact that the equality (2.1) is valid for all

x,y,z 2 X, the equality (2.1) directly yields the equality

~�ðx; u; zÞ ¼ ~�ðx�1; z; uÞ:Therefore we obtain that

~�ðx; y; zÞ � Eðy; uÞ 6 ~�ðx�1; z; uÞ ¼ ~�ðx; u; zÞ;i.e. ~� is transitive of the second order. Similar to (2.1), we easily see

that

~�ðx; y; zÞ ¼ ~�ðy; y�1; eÞ � ~�ðx; e; xÞ � ~�ðx; y; zÞ � ~�ðz; y�1; z � y�1Þ6 Eðz � y�1; xÞ ¼ ~�ðz; y�1; xÞ

and

~�ðz; y�1; xÞ ¼ ~�ðy�1; y; eÞ � ~�ðz; e; zÞ � ~�ðz; y�1; xÞ � ~�ðx; y; x � yÞ6 Eðx � y; zÞ ¼ ~�ðx; y; zÞ;

i.e.

~�ðx; y; zÞ ¼ ~�ðz; y�1; xÞ: ð2:2ÞSince (2.2) is true for all x,y,z 2 X, the equality (2.2) directly gives

the equality

~�ðu; y; zÞ ¼ ~�ðz; y�1; uÞ:Hence considering (2.2), we have

~�ðx; y; zÞ � Eðx; uÞ ¼ ~�ðz; y�1; xÞ � Eðx; uÞ 6 ~�ðz; y�1; uÞ ¼ ~�ðu; y; zÞ;

so the third order transitivity of ~� is got.

(vii) It is obvious from [3, Theorem 2.3] that ~� 6 vagð�Þ. The required

inclusions can be easily obtained from this fact and Definition 2.1.

(viii) If ~� is a perfect M-vague binary operation, i.e. ðX ; ~�Þ is a perfectM-vague group, then by (vi), ~� is transitive of the second order.

Therefore the assertion is straightforward from [1, Proposition

2.14(ii)]. Consider the case that E is regular w.r.t. D for some

D 2 ORDð~�Þ. Then the properties (v) and (vi) entail that

(X,vag(D)) is a perfect M-vague group w.r.t. P and E, and vag(D)

is transitive of the second order. Therefore, we have E(x,y) =

E(u,v) for all u 2 Inv(X,vag(D))(x,e), v 2 Inv(X,vag(D))(y,e) and e 2Id*(X,vag(D)). Using (vii), it is clear that Id�ðX ; ~�Þ � Id�

ðX ; vagðDÞÞ; InvðX ;~�Þðx; eÞ � InvðX ;vagðDÞÞðx; eÞ and InvðX ;~�Þ ðy; eÞ �InvðX ;vagðDÞÞðy; eÞ for all e 2 Id�ðX ; ~�Þ. Thus, we obtain that

M. Demirci / Information Sciences 176 (2006) 1488–1530 1493

E(x,y) = E(u,v) for all u 2 InvðX ;~�Þðx; eÞ; v 2 InvðX ;~�Þðy; eÞ and

e 2 Id�ðX ; ~�Þ. h

In this section, we introduce vague integral powers of elements in vague

groups and study their fundamental properties. For this purpose, it is necessary

to consider two essential components of the notion of the vague integral power

of elements called exact integral powers and successive integral powers of ele-

ments. In an analogous manner to exact products of elements in vague groups

[3,4], exact integral powers of elements in a vague group ðX ; ~�Þ are defined in

terms of ordinary descriptions of ~�:

Definition 2.3. Let ðX ; ~�Þ be an M-vague semigroup w.r.t. P and E, and for

n 2 Nþ, let f�iji ¼ 1; . . . ; n� 1g � ORDð~�Þ and a,a1,a2, . . . ,an 2 X.

(i) If a1 = a2 = � � � = an = a, thenQn

i¼1ai

� �Rf�iji¼1;...;n�1g

Qni¼1ai

� �Lf�iji¼1;...;n�1g

� �is

called the exact right (left) nth power of a 2 X w.r.t. {�iji = 1, . . . ,n � 1},

and is denoted by ½an�Rf�i ji¼1;...;n�1g ½an�Lf�i ji¼1;...;n�1g

� �. If ½an�Rf�iji¼1;...;n�1g coin-

cides with ½an�Lf�iji¼1;...;n�1g, then it is denoted by ½an�f�iji¼1;...;n�1g and called

the exact nth power of a 2 X w.r.t. {�iji = 1, . . . ,n � 1}. Furthermore,for D 2 ORDð~�Þ and for all i = 1, . . . ,n � 1, if �i = D, ½an�f�iji¼1;...;n�1g simply

denoted by [an]D.

(ii) If ðX ; ~�Þ is an M-vague group, then for a given inverse b of a w.r.t.

an identity element e of ðX ; ~�Þ; ½bn�Rf�i ji¼1;...;n�1g ½bn�Lf�iji¼1;...;n�1g

� �is called

the exact right (left) (�n)th power of a 2 X w.r.t. f�iji ¼ 1; . . . ; n� 1gand ðe; bÞ, and is represented by ½a�n�Rf�i ji¼1;...;n�1g

� �ðe;bÞ

½a�n�Lf�i ji¼1;...;n�1g

� �ðe;bÞ

� �. If ½a�n�Rf�iji¼1;...;n�1g

� �ðe;bÞ

coincides with

½a�n�Lf�i ji¼1;...;n�1g

� �ðe;bÞ

then it is denoted by ½a�n�f�iji¼1;...;n�1g

� �ðe;bÞ

called

the exact (�n)th power of a 2 X w.r.t {�iji = 1, . . . ,n � 1} and (e,b).

An exact (right or left) nth power of a in an M-vague semigroup ðX ; ~�Þ can

be comprehended as the formulation of the notion of nth power of a in ðX ; ~�Þby means of only ordinary descriptions of ~�. It should be noted here that thenotion of exact (right or left) nth power of a in ðX ; ~�Þ does not depend on

the shape of ~�, but just ordinary descriptions of ~�, which are certainly defined

binary operations.

In Definition 2.3, ½an�Rf�iji¼1;...;n�1g and ½an�Lf�iji¼1;...;n�1g are two elements of X that

can be explicitly written as

½an�Rf�i ji¼1;...;n�1g ¼ ð. . . ðða�1aÞ�2a . . .Þ�n�1aÞ

1494 M. Demirci / Information Sciences 176 (2006) 1488–1530

and

½an�Lf�i ji¼1;...;n�1g ¼ a�n�1ða�n�2ða�n�3ð. . . ða�2ða�1aÞÞ . . .ÞÞÞ:

If E is particularly taken as an M-equality on X, then ~� has a unique ordinary

description �, i.e. ORDð~�Þ ¼ f�g and the ordered pair (X,�) defines a semi-

group in the classical sense (see [1, Theorem 2.10(vi)]). In this case, since

�1 = �2 = � � � = �n�1 = � for all f�iji ¼ 1; . . . ; n� 1g � ORDð~�Þ, and by the factthat (X,�) forms a semigroup in the classical sense, we observe that

½an�Rf�i ji¼1;...;n�1g ¼ ½an�Lf�i ji¼1;...;n�1g ¼ ½an�f�i ji¼1;...;n�1g ¼ ½an��:

Furthermore, [an]� is nothing but the nth power an of a in (X,�) in the usual

sense. Given a semigroup (X,�) in the classical sense, since the notions of exact

right and left nth power of a in (X,�) are the same things, consideration of both

of them is completely unnecessary, and it is enough to consider the exact nth

power of a in (X,�), or simply an. In contrast to the classical situation, these

two notions do not coincide with each other in an M-vague semigroup ðX ; ~�Þin general. More clearly, if E is not an M-equality but just an M-equivalence

relation, then ½an�Rf�iji¼1;...;n�1g is not necessarily equal to ½an�Lf�iji¼1;...;n�1g in general,

but they relate to each other by the equivalence relation ½an�Rf�i ji¼1;...;n�1g�E½an�Lf�iji¼1;...;n�1g (cf. [3, Proposition 3.2]).

The notion of successive integral power of elements is another component of

the notion of vague integral power of elements and defined as a particular case

of the notion of successive product of elements.

Definition 2.4. Let ðX ; ~�Þ be an M-vague semigroup w.r.t. P and E. For a 2 Xand n 2 Nþ let the maps ½WR

~� ðanÞ�; ½WL~� ðanÞ� : X n ! L be defined by

½WR~� ðanÞ�ðu1; . . . ; unÞ ¼ WR

~� ½ða; a; . . . ; aÞ; ðu1; . . . ; unÞ�and

½WL~� ðanÞðu1; . . . ; unÞ ¼ WL

~� ½ða; a; . . . ; aÞ; ðu1; . . . ; unÞ�:

(i) ½WRðanÞ�ðu ; . . . ; u Þ ð½WLðanÞ�ðu ; . . . ; u ÞÞ is said to be the degree for

~� 1 n ~� 1 n

which (u1, . . . ,un) 2 Xn is a right (left) nth successive power of a.

(ii) (u1, . . . ,un) 2 Xn is called an exact right (left) nth successive power of a (orsimply, a right (left) nth successive power of a) iff u1 = a (un = a) and

½WR~� ðanÞ�ðu1; . . . ; unÞ ¼ 1 ð½WL

~� ðanÞ�ðu1; . . . ; unÞ ¼ 1Þ.(iii) Let ðX ; ~�Þ be an M-vague group w.r.t. P and E. For e 2 Id�ðX ; ~�Þ and

b 2 InvðX ;~�Þða; eÞ, the map ½WR~� ðb

nÞ� : X n ! Lð½WL~� ðb

n�:Xn! L) is denoted

by ½WR~� ða�nÞ�ðe;bÞð½WL

~� ða�nÞ�ðe;bÞÞ, and the element ½WR~� ða�nÞ�ðe;bÞðu1; . . . ; unÞ

ð½WL~� ða�nÞ�ðe;bÞðu1; . . . ; unÞÞ of L is said to be the degree for which

(u1, . . . ,un) 2 Xn is a right (left) (�n)th successive power of a w.r.t (e,b).

M. Demirci / Information Sciences 176 (2006) 1488–1530 1495

If there is no danger of confusion, we usually disregard the lower script ~� in

the mappings ½WR~� ðanÞ� and ½WL

~� ðanÞ� for simplicity.

For (u1, . . . ,un) 2 Xn, the elements ½WRðanÞ�ðu1; . . . ; unÞ and ½WLðanÞ�ðu1; . . . ; unÞ of L can be given in a more explicit form:

½WRðanÞ�ðu1; . . . ; unÞ ¼�n�1

i¼1 ~�ðui; a; uiþ1Þ; if u1 ¼ a

0; if u1 6¼ a

� ;

½WLðanÞ�ðu1; . . . ; unÞ ¼�n�1

i¼1 ~�ða; uiþ1; ui; Þ; if un ¼ a

0; if un 6¼ a

� for n P 2, and

½WRðaÞ�ðuÞ ¼ ½WLðaÞ�ðuÞ ¼1; if u ¼ a

0; if u 6¼ a

� for n = 1.

In the bivalent logic, i.e. L = {0,1} and * = ^, if the M-equivalence relationsP and E are particularly chosen as the crisp M-equalities on X · X and X, i.e.

for all x, x 0,y, y 0 2 X

P ððx;yÞ;ðx0;y0ÞÞ ¼1; ifðx;yÞ¼ ðx0;y0Þ0; otherwise

� and Eðx;yÞ¼

1; if x¼ y

0; otherwise

� ;

then an M-vague semigroup ðX ; ~�Þ corresponds to a semigroup (X,�) in the

classical sense in a one-to-one way, where ~� is given by ~�ðx; y; zÞ ¼1; if z ¼ x � y0; otherwise

� 8x; x0; y; y 0 2 X : In this special case, ½WRðanÞ�ðu1; . . . ; unÞ

ð½WLðanÞ�ðu1; . . . ; unÞÞ represents the truth value of the sentence

u1 ¼ a and u2 ¼ u1 � a and � � � and un ¼ un�1 � a

ðun ¼ a and un�1 ¼ a � un and � � � and u1 ¼ a � u2Þð2:3Þ

A right (left) nth successive power (u1, . . . ,un) 2 Xn of a is here nothing but an

n-tuple (u1, . . . ,un) making the sentence (2.3) true. Thus, in the general case,

½WRðanÞ�ðu1; . . . ; unÞ ð½WLðanÞ�ðu1; . . . ; unÞÞ can be conceived as the truth value

of a many- valued counterpart to (2.3). And a right (left) nth successive power

(u1, . . . ,un) 2 Xn of a corresponds to the certainly true case of this many-valued

counterpart to (2.3). For this reason, ½WRðanÞ�ðu1; . . . ; unÞ ð½WLðanÞ�ðu1; . . . ; unÞÞ can also be interpreted as the degree of (u1, . . . ,un) 2 Xn being a

right (left) nth successive power of a. Therewith, the mapping½WRðanÞ� ð½WLðanÞ�Þ can be thought of as the fuzzy set of all right (left) nth suc-

cessive powers of a.

In the case of bivalent logic, there exists a one-to-one connection between

integral powers of elements and exact successive integral powers of elements

1496 M. Demirci / Information Sciences 176 (2006) 1488–1530

in the sense that (u1, . . . ,un) 2 Xn is a right (left) nth successive power of a iff

uk = ak (uk = an�k+1) for all k = 1, . . . ,n. An analogous one-to-one connection

between exact integral powers of elements and exact successive integral powers

of elements in vague semigroups can be established in the following manner:

Proposition 2.5. Let ðX ; ~�Þ be an M-vague semigroup w.r.t. P and E, and a,u1, . . . , un 2 X for n P 2.

(i) (u1, . . . ,un) is a right (left) nth successive power of a iff there exists

f�iji ¼ 1; . . . ; n� 1g � ORDð~�Þ such that

uk ¼ ½ak�Rf�i ji¼1;...;k�1g ðuk ¼ ½an�kþ1�Lf�iji¼1;...;n�kgÞ

for all k = 1, . . . ,n.

(ii) If ðX ; ~�Þ is an M-vague group w.r.t. P and E, then for e 2 Id�ðX ; ~�Þ and

b 2 InuðX ;~�Þða; eÞ, (u1, . . . , un) is a right (left) (�n)th successive power of aw.r.t. (e,b) iff there exists f�iji ¼ 1; . . . ; n� 1g � ORDð~�Þ such that

uk ¼ ½a�k�Rf�iji¼1;...;k�1g

� �ðe;bÞ

uk ¼ ½a�ðn�kþ1Þ�Lf�iji¼1;...;n�kg

� �e;b

� �for all k = 1, . . . ,n.

Proof. Both the properties (i) and (ii) are direct consequences of [3, Proposi-

tion 3.4]. h

By taking into account the equalities (3.1) and (3.2) in Part I, and using Def-

inition 2.4, the connection between exact integral powers of elements and exactsuccessive integral powers of elements stated in Proposition 2.5 can be ex-

tended to exact integral powers of elements and the mapping ½WRðanÞ�ð½WLðanÞ�Þ: for n P 2, a,x 2 X and f�iji ¼ 1; . . . ; n� 2g � ORDð~�Þ, the degree

½WRðanÞ� a; ½a2�Rf�1g; ½a3�Rf�i ji¼1;2g; . . . ; ½an�1�Rf�i ji¼1;...;n�2g; x� �

½WLðanÞ� x; ½an�1�Lf�iji¼1;...;n�2g; . . . ; ½a3�Lf�i ji¼1;2g; ½a2�Lf�1g; a� �� �

of

a; ½a2�Rf�1g; ½a3�Rf�iji¼1;2g; . . . ; ½an�1�Rf�i ji¼1;...;n�2g; x� �

2 X n

x; ½an�1�Lf�iji¼1;...;n�2g; . . . ; ½a3�Lf�i ji¼1;2g; ½a2�Lf�1g; a� �

2 X n� �

being a right (left) nth successive power of a can be calculated in terms of

½an�1�Rf�i ji¼1;...;n�2g ð½an�1�Lf�i ji¼1;...;n�2gÞ by the equality

M. Demirci / Information Sciences 176 (2006) 1488–1530 1497

½WRðanÞ� a; ½a2�Rf�1g; ½a3�Rf�i ji¼1;2g; . . . ; ½an�1�Rf�iji¼1;...;n�2g; x� �

¼~� ½an�1�Rf�i ji¼1;...;n�2g; a; x� �½WLðanÞ�ðx; ½an�1�Lf�iji¼1;...;n�2g; . . . ; ½a3�Lf�i ji¼1;2g; . . . ; ½a2�Lf�1g; aÞ�

¼~� a; ½an�1�Lf�i ji¼1;...;n�2g; x� ��

:

ð2:4Þ

Definition 2.6. Let ðX ; ~�Þ be an M-vague semigroup w.r.t. P and E.

(i) For a 2 X, n 2 Nþ and a1 = a2 = � � � = an = a, the L-fuzzy sets,

PwRðanÞ; PwLðanÞ : X ! L, defined by

½PwRðanÞ�ðxÞ ¼ fYR

ða1; a2; . . . ; anÞ �

ðxÞ

and

½PwLðanÞ�ðxÞ ¼ fYL

ða1; a2; . . . ; anÞ �

ðxÞ

for all x 2 X, are called the right vague nth power of a and the left vague

nth power of a, respectively. If PwRðanÞ coincides with PwLðanÞ, then it is

called the vague nth power of a and represented by PwðanÞ.(ii) If ðX ; ~�Þ is an M-vague group, then for e 2 Id�ðX ; ~�Þ and b 2 InvðX ;~�Þða; eÞ,

PwRðbnÞ ðPwLðbnÞÞ is called the right (left) vague (�n)th power of a w.r.t.

(e,b), and is denoted by ½PwRða�nÞ�ðe;bÞ ðPwLða�nÞ�ðe;bÞÞ.

Furthermore, if ½PwRða�nÞ�ðe;bÞ coincides with ½PwLða�nÞ�ðe;bÞ, then it is

called the vague (�n)th power of a w.r.t. (e,b) and denoted by

½Pwða�nÞ�ðe;bÞ.

For the sake of completion of the terminology in Part I, if the vague binary

operation ~� in Definition 2.6 is denoted by an additive notation ~þ, the right vague

nth power of a, the left vague nth power of a and the vague nth power of a will be

represented by PwRðnaÞ; PwLðnaÞ and Pw(na), and they are called the right

vague nth multiple of a, the left vague nth multiple of a and the vague nth multipleof a, respectively. Similarly, for an additive notation ~þ, the word ‘‘power’’ and

the product notation ‘‘�’’ in Definition 2.3 are replaced by ‘‘multiple’’ and the

sum notation ‘‘P

’’, respectively. Throughout this paper, a vague binary opera-

tion ~� always stands for a multiplicative notation, unless otherwise mentioned.

In order to clarify the role of exact integral powers and successive integral

powers of elements in the definition of vague integral powers of elements, it

is useful to note that the right and left vague nth powers of a can be explicitly

expressed as

1498 M. Demirci / Information Sciences 176 (2006) 1488–1530

½PwRðanÞ�ðxÞ¼_

~�ð½an�1�Rf�i ji¼1;...;n�2g;a;xÞjf�iji¼1; . . . ;n�2g�ORDð~�Þn o

¼_n

WRðanÞ� �

ða;v2;v3; . . . ;vn�1;xÞ

vk¼½ak�Rf�i ji¼1;...;k�1g;k¼2; . . . ;n�1;f�iji¼1; . . . ;n�2g�ORDð~�Þ��� o

and

½PwLðanÞ�ðxÞ¼_

~� a; an�1� �L

f�i ji¼1;...;n�2g;a;x� �

jf�iji¼1; . . . ;n�2g�ORDð~�Þn o

¼_n

WLðanÞ� �

ðx;v2;v3; . . . ;vn�1;aÞ

vn�k¼½akþ1�Lf�i ji¼1;...;kg; k¼1; . . . ;n�2;f�iji¼1; . . . ;n�2g�ORDð~�Þ��� o

for n P 2, and

½PwRðanÞ�ðxÞ ¼ ½PwLðanÞ�ðxÞ ¼1; if x ¼ a

0; if x 6¼ a

� for n = 1. Taking into consideration these explicit expressions of PwRðanÞ and

PwLðanÞ, we easily note that for all f�iji ¼ 1; . . . ; n� 1g � ORDð~�Þ,

½PwRðanÞ�ð½an�Rf�iji¼1;...;n�1gÞ ¼ ½PwLðanÞ�ð½an�Lf�iji¼1;...;n�1gÞ ¼ 1:

Because of this observation, PwRðanÞ ðPwLðanÞÞ can be thought of as the fuzzy

set of all exact right (left) nth powers of a. Second, ½PwRðanÞ�ðxÞ ð½PwLðanÞ�ðxÞÞcan be interpreted to be the degree for which x 2 X is an exact right (left) nth

power of a. As a comparison of exact right (left) nth powers of a and the right

(left) vague nth power of a, the letter depends on the shape of the underlying

M-vague binary operation ~�, and carries the information on the measure of

a given element x 2 X how much it is an exact right (left) nth power of a,

although the former is formulated by means of ordinary descriptions of ~�,and does not provide any further information.

If we denote the set of all exact right (left) nth powers of a by

CRðaÞ ðCLðaÞÞ, i.e.

CRðaÞ ¼ ½an�Rf�i ji¼1;...;n�1gjf�iji ¼ 1; . . . ; n� 1g � ORDð~�Þn o

CLðaÞ ¼ ½an�Lf�i ji¼1;...;n�1gjf�iji ¼ 1; . . . ; n� 1g � ORDð~�Þn o� �

;

then it is evident that CRðaÞ ðCLðaÞÞ is a subset of kerðPwRðanÞÞðkerðPwLðanÞÞÞ, where kerðPwRðanÞÞ ðkerðPwLðanÞÞÞ denotes the kernel ofPwRðanÞðPwLðanÞÞ, i.e. kerðPwRðanÞÞ ¼ fx 2 X j½PwRðanÞ�ðxÞ ¼ 1g ðkerðPwL

ðanÞÞ ¼ fx 2 X j½PwLðanÞÞ�ðxÞ ¼ 1gÞ. For this reason, PwRðanÞ ðPwLðanÞÞ is a

natural fuzzification of CRðaÞ ðCLðaÞÞ. Moreover, if the underlying M-vague

binary operation ~� on X is perfect, then CRðaÞ; CLðaÞ, kerðPwRðanÞÞ and

M. Demirci / Information Sciences 176 (2006) 1488–1530 1499

kerðPwLðanÞÞ become the same thing, which is simply given by the equiva-

lence class of any exact right (left) nth power of a w.r.t. �E (cf. [4, Theorem

4.7(iii)]).

For a given semigroup (X,�) in the classical sense, for a, b 2 X and n 2 Nþ,

the implication ‘‘if a = b, then an = bn’’ is a well-known fact in classical alge-

bra. Now we show how this simple fact can be extended to vague semi-groups:

Proposition 2.7. Let M = (L,6,*) denote an integral, commutative cl-monoid

and let ðX ; ~�Þ be an M-vague semigroup w.r.t. P and E with ~� transitive of both

the second and third order. Then for n 2 Nþ and for all a, b, x 2 X,

(i) ½PwRðanÞ�ðxÞ � ½Eða; bÞ�n 6 ½PwRðbnÞ�ðxÞ,(ii) ½PwLðanÞ�ðxÞ � ½Eða; bÞ�n 6 ½PwLðbnÞ�ðxÞ.

In order to prove this result, we need the following lemma:

Lemma 2.8. Let ðX ; ~�Þ be an M-vague semigroup w.r.t. P and E such that ~� istransitive of the both second and third order. Then for n 2 Nþ and for all

a,b,w2, . . . ,wn 2 X, the following relations hold:

(i) ½WRðanÞ�ða;w2; . . . ;wnÞ � ðEða; bÞÞn 6 ½WRðbnÞ�ðb;w2; . . . ;wnÞ,(ii) ½WLðanÞ�ðw2; . . . ;wn; aÞ � ðEða; bÞÞn 6 ½WLðbnÞ�ðw2; . . . ;wn; bÞ.

Proof. (i) For n = 1, the assertion is clear. For n P 2, we have

½WRðanÞ�ða;w2;...;wnÞ�ðEða;bÞÞn¼½~�ða;a;w2Þ�Eða;bÞ�Eða;bÞ�

�ð½~�ðw2;a;w3Þ�Eða;bÞ�� ����½~�ðwn�1;a;wnÞ�Eða;bÞ�Þ

6 ½~�ðb;a;w2Þ�Eða;bÞ��ð½~�ðw2;a;w3Þ�Eða;bÞ������ ½~�ðwn�1;a;wnÞ�Eða;bÞ�Þ

6 ½~�ðb;b;w2Þ��ð½~�ðw2;b;w3Þ����� ~�ðwn�1;b;wnÞ¼½WRðbnÞ�ðb;w2;...;wnÞÞ:

(ii) is analogous to (i). h

Proof of Proposition 2.7. (i) For n = 1, the assertion is clear. Consider the case

n P 2. For {�iji = 1, . . . ,n � 1}, fDiji ¼ 1; . . . ; n� 1g � ORDð~�Þ, let us choose

the elements w2, . . . ,wn 2 X in Lemma 2.8(i) such that wk ¼ ½ak�Rf�iji¼1;...;k�1gfor all k = 2, . . . ,n, and consider the equality (2.4). Then, by Lemma 2.8(i),

and using the third-order transitivity of ~�, we estimate the followinginequalities:

1500 M. Demirci / Information Sciences 176 (2006) 1488–1530

½Eða; bÞ�n ¼ ½WRðanÞ�ða; ½a2�Rf�1g; ½a3�Rf�iji¼1;2g; . . . ; ½an�Rf�i ji¼1;...;n�1gÞ � ½Eða; bÞ�n

6 ½WRðbnÞ�ðb; ½a2�Rf�1g; ½a3�Rf�iji¼1;2g; . . . ; ½an�Rf�iji¼1;...;n�1gÞ

¼ ~�ðb; b; ½a2�Rf�1gÞ � ~�ð½a2�Rf�1g; b; ½a3�Rf�iji¼1;2gÞ � � � �

� ~�ð½an�1�Rf�iji¼1;...;n�2g; b; ½an�Rf�i ji¼1;...;n�1gÞ

¼ ~�ðb; b; ½a2�Rf�1gÞ � ~�ðb; b; ½b2�RfD1gÞ� � ~�ð½a2�Rf�1g; b; ½a3�Rf�iji¼1;2gÞ � � � �

� ~�ð½an�1�Rf�iji¼1;...;n�2g; b; ½an�Rf�i ji¼1;...;n�1gÞ

6 Eð½a2�Rf�1g; ½b2�RfD1gÞ � ~�ð½a2�Rf�1g; b; ½a3�Rf�iji¼1;2gÞ � � � �

� ~�ð½an�1�Rf�iji¼1;...;n�2g; b; ½an�Rf�i ji¼1;...;n�1gÞ

6 ~�ð½b2�RfD1g; b; ½a3�Rf�iji¼1;2gÞ � � � �

� ~�ð½an�1�Rf�iji¼1;...;n�2g; b; ½an�Rf�i ji¼1;...;n�1gÞ

¼ ~�ð½b2�RfD1g; b; ½a3�Rf�i ji¼1;2gÞ � ~�ð½b2�RfD1g; b; ½b3�RfDiji¼1;2gÞ � � � �

� ~�ð½an�1�Rf�iji¼1;...;n�2g; b; ½an�Rf�i ji¼1;...;n�1gÞ

6 Eð½a3�Rf�iji¼1;2g; ½b3�RfDi ji¼1;2gÞ � ~�ð½a3�Rf�i ji¼1;2g; b; ½a4�Rf�i ji¼1;2;3gÞ � � � �

� ~�ð½an�1�Rf�iji¼1;...;n�2g; b; ½an�Rf�i ji¼1;...;n�1gÞ

6 � � � 6 Eð½an�Rf�iji¼1;...;n�1g; ½bn�RfDi ji¼1;...;n�1gÞ:

Thus, we get the inequality

½Eða; bÞ�n 6 Eð½an�Rf�iji¼1;...;n�1g; ½bn�RfDi ji¼1;...;n�1gÞ: ð2:5Þ

Then, utilizing the inequality (2.5), we can write

~�ð½an�1�Rf�iji¼1;...;n�2g;a;xÞ�½Eða;bÞ�n

¼ ~�ð½an�1�Rf�iji¼1;...;n�2g;a;xÞ�½Eða;bÞ�n�1�Eða;bÞ

6 ~�ð½an�1�Rf�iji¼1;...;n�2g;a;xÞ�Eð½an�1�Rf�i ji¼1;...;n�2g;½bn�1�RfDiji¼1;...;n�2gÞ�Eða;bÞ

6 ~�ð½bn�1�RfDiji¼1;...;n�2g;a;xÞ�Eða;bÞ6 ~�ð½bn�1�RfDiji¼1;...;n�2g;b;xÞ6 ½PwRðbnÞ�ðxÞ

and hence we obtain

~�ð½an�1�Rf�iji¼1;...;n�2g; a; xÞ � ½Eða; bÞ�n6 ½PwRðbnÞ�ðxÞ:

Because of the distributivity of * over arbitrary joins, this inequality entails

that

M. Demirci / Information Sciences 176 (2006) 1488–1530 1501

½PwRðanÞ�ðxÞ � ½Eða;bÞ�n

¼_f~�ð½an�1�Rf�i ji¼1;...;n�2g;a;xÞjf�iji¼ 1; . . . ;n� 2g ORDð~�Þg � ½Eða;bÞ�n

¼_f~�ð½an�1�Rf�i ji¼1;...;n�2g;a;xÞ � ½Eða;bÞ�njf�iji¼ 1; . . . ;n� 2g

�ORDð~�Þg6 ½PwRðbnÞ�ðxÞ:

In a similar way, the property (ii) can be easily shown by applying Lemma2.8(ii). h

As an immediate consequence of Proposition 2.7, it is easy to see that under the

hypothesis of Proposition 2.7, if a,b 2 X belong to the same equivalence class

w.r.t. �E, i.e. a �E b, then ½PwRðanÞ�ðxÞ ¼ ½PwRðbnÞ�ðxÞ 8x 2 X , i.e. PwRðanÞcoincides with PwRðbnÞ. The same result will clearly also be valid for PwLðanÞand PwLðbnÞ. As another consequence of Proposition 2.7, if the underlying

integral, commutative cl-monoid M = (L,6, *) in Proposition 2.7 is particu-

larly chosen as ([0,1],6,^), then we easily obtain the equalities

½PwRðanÞ�ðxÞ ^ Eða; bÞ ¼ ½PwRðbnÞ�ðxÞ ^ Eða; bÞand

½PwLðanÞ�ðxÞ ^ Eða; bÞ ¼ ½PwLðbnÞ�ðxÞ ^ Eða; bÞ:Due to the fact that vague integral powers of elements are defined as a special

case of vague products, they preserve various elementary properties of vague

products:

Proposition 2.9. Let ðX ; ~�Þ be an M-vague semigroup w.r.t. P and E. For n P 2,

a 2 X, f�iji ¼ 1; . . . ; n� 1g ORDð~�Þ and U;V 2 fR;Lg, the followingstatements are true:

(i) PwUðanÞ relates to PwVðanÞ by the inequality

½PwUðanÞ�ðxÞ � ½PwVðanÞ�ðyÞ 6 Eðx; yÞ 8x; y 2 X :

(ii) If ~� transitive of the third (second) order, then the right (left) vague nth

power of a can be represented by

½PwRðanÞ�ðxÞ ¼ ~�ð½an�1�Uf�iji¼1;...;n�2g; a; xÞ;

ð½PwLðanÞ�ðxÞ ¼ ~�ða; ½an�1�Uf�i ji¼1;...;n�2g; xÞÞ 8x 2 X :

(iii) The right and left vague nth powers of a satisfy the inequalities

½PwRðanÞ�ðxÞ 6 Eð½an�Uf�iji¼1;...;n�1g; xÞ

and

½PwLðanÞ�ðxÞ 6 Eð½an�Uf�i ji¼1;...;n�1g; xÞ

1502 M. Demirci / Information Sciences 176 (2006) 1488–1530

for all x 2 X, and the equalities hold if ~� is a perfect M-vague binary

operation.

(iv) If ~� is a perfect M-vague binary operation having an associative ordinary

description D, then PwRðanÞ coincides with PwLðanÞ, and the vague nth

power Pw(an) of a is given by

½PwðanÞ�ðxÞ ¼ Eð½an�D; xÞ 8x 2 X : ð2:6Þ

Proof. The required properties are direct results of [3, Propositions 3.6 and

3.7]. h

Remark 2.10

(i) If ðX ; ~�Þ defines a perfect M-vague semigroup w.r.t. P and E, and if E is

an M-equality on X, then for D ¼ ordð~�Þ, since D is an associative crisp

binary operation on X (see [1, Theorem 2.10(vi)]) it follows from Propo-

sition 2.9(iv) that for n P 2 the vague nth power Pw(an) of a is explicitly

formulated by the equality (2.6).(ii) Let ER be a regular M-equivalence relation on R w.r.t. the usual addition

(multiplication) operation ‘‘+’’ (‘‘Æ’’). For an M-equivalence relation ER2

on R2, let us consider a perfect M-vague addition (multiplication) opera-

tion ~� on R w.r.t. ER2 and ER. By invoking [2, Corollary 2.15], we easily

see that ðR; ~�Þ forms a perfect M-vague semigroup w.r.t. ER2 and ER.

Then, since ‘‘+’’ (‘‘Æ’’) is an associative ordinary description of ~�, it is

an immediate consequence of Proposition 2.9(iv) that for n P 2 the vague

nth multiple (power) Pw(na) (Pw(an)) of a 2 R w.r.t. the perfect M-vagueaddition (multiplication) operation ~� is given by

½PwðnaÞ�ðxÞ ¼ ERðn � a; xÞ ð½PwðanÞ�ðxÞ ¼ ERðan; xÞÞ 8x 2 R:

Example 2.11

(i) For the particular integral, commutative cqm-lattice M = ([0,1],6,Lck),

where Lck denotes the Lukasiewicz�s t-norm, i.e. Lck(x,y) = max

{x + y � 1, 0} "x,y 2 [0,1], let us reconsider the M-equivalence relations

EðþÞR on R; EðþÞR2 on R2 and the perfect M-vague addition operation ~þ

EðþÞR

on R w.r.t. EðþÞR2 and EðþÞR , introduced in [3, Example 3.9(i)]. Then for

n P 2, the vague nth multiple Pw(na) of a 2 R w.r.t. ~þEðþÞ

R

is explicitly

given by

½PwðnaÞ�ðxÞ ¼ EðþÞR ðn � a; xÞ ¼ 1� minfjn � a� xj; 1g 8x 2 R

M. Demirci / Information Sciences 176 (2006) 1488–1530 1503

(ii) For the product t-norm Pr(x,y) = xÆy "x, y 2 [0,1], let M = ([0, 1],6,Pr)

stand for the underlying integral, commutative cqm-lattice, and let

ER; ER2 and ~�ERdenote the M-equivalence relations and the perfect

M-vague multiplication operation, presented in [3, Example 3.9(ii)]. Then

for n P 2, the vague nth power Pw(an) of a 2 R w.r.t. ~�ERcan be calcu-

lated as

½PwðanÞ�ðxÞ ¼ ERðan; xÞ ¼ min an

x

�� ��; xan

�� �� �; if a; x 2 R� f0g

EcRðan; xÞ; otherwise

( )8x 2 R:

In a measurement process of a quantity q, because of the uncertainty result-

ing from discrete scales of measurement instruments, it is shown in [3] that the

indistinguishability of any two possible values of q can be naturally and con-

sistently modelled by indistinguishability operators. Thereby, for a measure-ment scale k, for M = ([0,1],6, *) and a suitably chosen M-equivalence

relation (or simply, an indistinguishability operator) EkR on R, the degree of

indistinguishability of any two measured values or possible values x,y of q

according to the scale k is given by the real number EkRðx; yÞ 2 ½0; 1�. Similarly,

for a suitable indistinguishability operator EkR2 : R2 R2 ! ½0; 1� on R2 and for

any possible values x; x0; y; y 0 2 R of q according to the scale k, the degree

of indistinguishability of ordered pairs (x,y) and (x 0,y 0) is given by

EkR2ððx; yÞ; ðx0; y 0ÞÞ. Because of these gradual representations of the indistin-

guishability of points of R and R2, it is natural to consider addition and mul-

tiplication operations as vaguely defined addition and multiplication

operations on R which are denoted by an M-vague addition operation ~þk

and an M-vague addition operation ~�k w.r.t. EkR2 and Ek

R. For the sake of sim-

plicity, we assume that ~þkð~�kÞ is a perfect M-vague addition (multiplication)

operation on R w.r.t. EkR2 and Ek

R. Then for any possible values

a1; a2; . . . ; an 2 R of the quantity q according to the scale k (n P 2), the sum

(product) of these values is calculated as the vague sum (product)fPkða1; a2; . . . ; anÞ ðfQkða1; a2; . . . ; anÞÞ of a1,a2, . . . ,an w.r.t ~þkð~�kÞ [3]:

gXkða1; a2; . . . ; anÞ

h iðxÞ ¼ Ek

R

Xn

i¼1

ai; x

!

fYkða1; a2; . . . ; anÞ

h iðxÞ ¼ Ek

R

Yn

i¼1

ai; x

! !8x 2 R:

As a result of this, for any possible value a 2 R of the quantity q according

to the scale k and for n P 2, the nth multiple (power) of a is computed as the

vague nth multiple (power) Pwk(na) (Pwk(an)) of a 2 R w.r.t. ~þk ð~�kÞ:½PwkðnaÞ�ðxÞ ¼ Ek

Rðn � a; xÞ ð½PwkðanÞ�ðxÞ ¼ EkRðan; xÞÞ 8x 2 R:

1504 M. Demirci / Information Sciences 176 (2006) 1488–1530

It is shown in [3] that given indistinguishability operators EkR and Ek

R2 and

the perfect M-vague addition (multiplication) operation ~þk ð~�kÞ on R according

to the scale k can be converted to indistinguishability operators EcR and Ec

R2 and

a perfect M-vague addition (multiplication) operation ~þc ð~�cÞ on R according

to the other scale c. With the help of this conversion, for any possible value

a 2 R of the quantity q obtained in the measurement process according tothe scale c and for n P 2, we compute the nth multiple (power) of a as the

vague nth multiple (power) Pwc(na) (Pwc(an)) of a 2 R w.r.t. ~þcð~�cÞ. In an ana-

logous manner to [3, Example 3.10], we show in the following example how

this computation can be done.

Example 2.12

(i) Let k and c be two interval scales [6], i.e. for some a, b 2 R with a > 0, theaffine transformation /int(x) = a Æx + b converts c to k [6]. Let us assume

that EkR, Ek

R2 and ~þkare, respectively, Lck-indistinguishability operators

EðþÞR and EðþÞR2 and the perfect M-vague addition operation on R w.r.t.

EðþÞR2 and EðþÞR , defined in [3, Example. 3.9(i)]. Then the maps Ec

R:R R! [0, 1], Ec

R2 : R2 R2 ! [0,1] and the fuzzy relation

~þc 2 ½0; 1�R3

, defined by

EcRðx; yÞ ¼ Ek

Rð/intðxÞ;/intðyÞÞ ¼ 1� minfa � jðx� yÞj; 1g;

EcR2ððx; yÞ; ðx0; y 0ÞÞ ¼ Ek

R2ðð/intðxÞ;/intðyÞÞ; ð/intðx0Þ;/intðy 0ÞÞÞ¼ 1� minfa � jðxþ yÞ � ðx0 þ y0Þj; 1g

and

~þcðx; y; zÞ ¼ EcRðxþ y; zÞ ¼ 1� minfa � jðxþ yÞ � zj; 1g

for all x, x0, y, y 0; z 2 R, are Lck-indistinguishability operators and a per-

fect M-vague addition operation on R w.r.t. EcR and Ec

R2 , respectively [3,

Example 3.10(i)]. Furthermore, since EcR is regular w.r.t. ‘‘+’’, and by Re-

mark 2.10(ii), for n P 2 and for any possible value a 2 R of q, the vague

nth multiple Pwc(na) of a w.r.t. ~þcis given by

½PwcðnaÞ�ðxÞ ¼ EcRðn � a; xÞ ¼ 1� minfa � jðn � aÞ � xj; 1g 8x 2 R:

(ii) Given two log-interval scales k and c [6], suppose that for some a, x 2 R

such that a > 0 and x > 0, their permissible transformation (power trans-

formation [6]) /log-int(x) = a Æ xx is defined on R and converts c to k. Let

EkR; Ek

R2 and ��kbe, respectively, the Pr-indistinguishability operators

ER; ER2 and the perfect M-vague multiplication operation ��ERw.r.t.

M. Demirci / Information Sciences 176 (2006) 1488–1530 1505

ER2 and ER defined in [3, Example 3.9(ii)]. It is shown in [3, Example

3.10(ii)] that the maps EcR : R R! ½0; 1�, Ec

R2 : R2 R2 ! ½0; 1� and

the fuzzy relation ��c 2 ½0; 1�R3

, defined by

EcRðx;yÞ¼Ek

Rð/log-intðxÞ;/log-intðyÞÞ¼min x

y

� �x��� ���; yx

� �x�� ��n o; if x;y2R�f0g

EcRðx;yÞ; otherwise

8<:9=;;

EcR2ððx; yÞ; ðx0; y 0ÞÞ ¼ Ek

R2ðð/log-intðxÞ;/log-intðyÞÞ; ð/log-intðx0Þ;/log-intðy 0ÞÞÞ

¼ EcRðx � y; x0 � y 0Þ

¼min x�y

x0 �y0

� �x��� ���; x0 �y0x�y

� �x��� ���n o; if x; x0; y; y0 2 R� f0g

EcR2ððx; yÞ; ðx0; y 0ÞÞ; otherwise

8<:9=;

and

��cðx;y;zÞ¼EcRðx � y;zÞ¼

min x�yz

� �x�� ��; zx�y

� �x��� ���n o; if x;y;z2R�f0g

1; if z¼ x � y¼ 0

0; otherwise

8>><>>:9>>=>>;

for all x, y, x0; y0 2 R, are Pr-indistinguishability operators and a perfect

M-vague multiplication operation on R w.r.t. EcR2 and Ec

R, respectively.

Furthermore, EcR is regular w.r.t. ‘‘ Æ ’’. Thus we obtain from Remark

2.10(ii) that for n P 2 and for any possible value a 2 R of q, the vague

nth power Pwc(an) of a w.r.t ��c

is given by

½PwcðanÞ�ðxÞ¼EcRðan;xÞ¼ min an

x

� �x�� ��; xan

� �x�� �� �; if a;x2R�f0g

EcRðan;xÞ; otherwise

( )8x2R:

The measurement of the perimeter of a square A can be considered as a simple

example for practical applications of Example 2.12(i). For this measurement,

we use two different rods with units of centimeter (cm) and millimeter (mm)

each as in the measurement of the perimeter of a triangle considered in [3].

Here the rod with the unit of cm (mm) corresponds to the scale k (c). The per-missible transformation converting mm readings to cm readings is given by

/ratioðxÞ ¼ x108x 2 R [3], i.e. k and c are ratio scales [6]. If we obtain the real

number a as the length of one of the sides of A according to the scale k (c),

and if we refer to Lck-indistinguishability operators EkR; Ek

R2 ðEcR; Ec

R2Þ and

the perfect M-vague addition operation ~þkð ~þcÞ considered in Example

2.12(i), then the perimeter of A in the unit of cm (mm) is calculated as the vague

fourth multiple of Pwk(4a) (Pwc(4a)) of a w.r.t. ~þkð ~þcÞ, given by

1506 M. Demirci / Information Sciences 176 (2006) 1488–1530

½Pwkð4aÞ�ðxÞ ¼ EkRð4 � a; xÞ ¼ 1� minfjð4 � aÞ � xj; 1g

½Pwcð4aÞ�ðxÞ ¼ EcRð4 � a; xÞ ¼ 1� min

1

10jð4 � aÞ � xj; 1

� � �8x 2 R.

Pwk(4a) and Pwc(4a) are two triangular fuzzy numbers centered at 4 Æa, and are

sketched in Figs. 1 and 2, respectively.

As an example for the realization of Example 2.12(ii), we can be interested in

the measurement of the volume of a cube. For the purpose of the side measure-

ment, we use the same rods (or simply the same scales) in the previous example,

and consider the Pr-indistinguishability operators EkRðE

cRÞ; Ek

R2 ðEcR2Þ and the

perfect M-vague multiplication operation ~�k (~�c) in Example 2.12(ii) for the side

measurement according to k (c). Then for the measured value a of the length ofits edges according to the scale k (c), the volume of the cube in the unit of cm3

(mm3) is computed as the vague third power Pwk(a3) (Pwc(a3)) of a w.r.t. ~�k (~�c),

given by

½Pwkða3Þ�ðxÞ ¼ ½Pwcða3Þ�ðxÞ ¼ EkRða3; xÞ ¼ min a3

jxj ;jxja3

n o; if x 6¼ 0

0; if x ¼ 0

( ):

for all x 2 R, where Pwk(a3) is represented in Fig. 3.

Fig. 1. The perimeter of the square in the unit of cm.

Fig. 2. The perimeter of the square in the unit of mm.

Fig. 3. The volume of the cube in the unit of cm3 (mm3).

M. Demirci / Information Sciences 176 (2006) 1488–1530 1507

It should be noted in Example 2.12 that if we restrict our attention to the

ratio scales k and c in both cases (i) and (ii) of Example 2.12, i.e. k and c possess

the permissible transformation /ratio : R! R defined by /ratio(x) = aÆx for

some a > 0 and for all x 2 R [6], which is also known as the similarity transfor-mation [6], then the vague nth multiples (powers) Pwk(na) and Pwc(na) (Pwk(an)

and Pwc(an)) of a possible value a of q satisfy the equality

½PwcðnaÞ�ðxÞ ¼ ½Pwkðn/ratioðaÞÞ�ð/ratioðxÞÞ

ð½PwcðanÞ�ðxÞ ¼ ½Pwkðð/ratioðaÞÞnÞ�ð/ratioðxÞÞÞ:

This equality can be thought of as the invariance of the vague nth multiple

(power) of a possible value of q under the ratio scales.

3. Application of the generalized vague associative law to integral powers of

elements in vague groups

In a semigroup (group) (X, o), for all m, n 2 Nþðm; n 2 ZÞ and a 2 X, the

equality an � am ¼ anþm is a well-known elementary property of integral powers

of elements in classical algebra. It is natural to expect an analogous result for

vague integral powers of elements in vague groups. In the theorem given below,

it is shown how this simple property can be carried to vague integral powers of

elements in vague groups.

Theorem 3.1. For an integral, commutative cl-monoid M ¼ ðL;6; �Þ, let (X ; ~�)denote an M-vague semigroup w.r.t. P and E, and D 2 ORDð~�Þ. Let one of the

conditions:

(a) ~� is transitive of both the first and third order,

(b) E is regular w.r.t. D

1508 M. Demirci / Information Sciences 176 (2006) 1488–1530

be satisfied.

(i) For m, n 2 Nþ and for all a, x, y, z 2 X,

½PwRðanÞ�ðxÞ � ½PwRðamÞ�ðyÞ � ½PwRðanþmÞ�ðzÞ 6 EðxDy; zÞ:Furthermore, if ðX ; ~�Þ forms an M-vague group w.r.t. P and E, then form; n 2 Nþ and for all a; x; y; z 2 X ; e 2 Id�ðX ; ~�Þ; b 2 InvðX ;~�Þða; eÞ, the fol-

lowing properties are valid:

(ii) ½PwRða�nÞ�ðe;bÞðxÞ � ½PwRða�mÞ�ðe;bÞðyÞ � ½PwRða�ðnþmÞÞ�ðe;bÞðzÞ 6 EðxDy; zÞ:

(iii) For m > n

½PwRðamÞ�ðxÞ � ½PwRða�nÞ�ðe;bÞðyÞ � ½PwRðaðm�nÞÞ�ðzÞ 6 EðxDy; zÞ:

(iv) For m = n

½PwRðanÞ�ðxÞ � ½PwRða�nÞ�ðe;bÞðyÞ 6 EðxDy; eÞ:

(v) For m < n

½PwRðamÞ�ðxÞ � ½PwRða�nÞ�ðe;bÞðyÞ � ½PwRða�ðn�mÞÞ�ðe;bÞðzÞ 6 EðxDy; zÞ:

The proof requires the following lemmas:

Lemma 3.2. Let (X ; ~�) be an M-vague group w.r.t. P and E, and D 2 ORDð~�Þ.Assume that one of the following two conditions is satisfied.

(a) ~� is a perfect M-vague binary operation,

(b) E is regular w.r.t. D.

Then for m; n 2 Nþ, a,u1, . . . , um,v�1, . . . , v�n 2 X, e 2 Id�ðX ; ~�Þ and b 2 InvðX ;~�Þða; eÞ, next properties are valid:

(i) If m > n

ð½WRðamÞ�Þðu1; . . . ; umÞÞ2 � ½WRða�nÞ�ðe;bÞðv�1; . . . ; v�nÞ

6 ½WRðam�nÞ�ðu1; . . . ; um�nÞ � EðumDv�n; um�nÞ:

(ii) If m = n

ð½WRðanÞ�ðu1; . . . ; unÞÞ2 � ½WRða�nÞ�ðe;bÞðv�1; . . . ; v�nÞ 6 EðunDv�n; eÞ:

(iii) If m < n

ð½WRðamÞ�ðu1; . . . ; umÞÞ2 � ½WRða�nÞ�ðe;bÞðv�1; . . . ; v�nÞ� ½WRða�ðn�mÞÞ�ðe;bÞðv�1; . . . ; v�ðn�mÞÞ 6 EðumDv�n; v�ðn�mÞÞ:

M. Demirci / Information Sciences 176 (2006) 1488–1530 1509

Lemma 3.3. For a given M-vague group ðX ; ~�Þ w.r.t. P and E and for

D 2 ORDð~�Þ, let one of the conditions (a) or (b) in Lemma 3.2 be true. Then for

m; n 2 Nþ; a 2 X and f�iji ¼ 1; . . . ;m� 1g; fDiji ¼ 1; . . . ; n� 1g � ORDð~�Þ,the following properties are true:

(i) For m > n

½am�Rf�iji¼1;...;m�1gD ½a�n�RfDiji¼1;...;n�1g

� �ðe;bÞ

� ��E½am�n�Rf�i ji¼1;...;m�n�1g:

(ii) For m = n

½an�Rf�iji¼1;...;n�1gD ½a�n�RfDiji¼1;...;n�1g

� �ðe;bÞ

� ��Ee:

(iii) For m < n

½am�Rf�iji¼1;...;m�1gD ½a�n�RfDiji¼1;...;n�1g

� �ðe;bÞ

� ��Eð½a�ðn�mÞ�RfDiji¼1;...;n�m�1gÞðe;bÞ:

Proof. (i) Let us first choose the elements u1, . . .,um, v�1, . . .,v�n 2 X in Lemma

3.2(i) such that uk ¼ ½ak�Rf�i ji¼1;...;k�1g and v�l ¼ ½a�l�RfDiji¼1;...;l�1g

� �ðe;bÞ

for all k =

1, . . . ,m and l = 1, . . . ,n. Then, by Proposition 2.5(i) and (ii), we have

½WRðamÞ�ðu1; . . . ; umÞ ¼ ½WRðam�nÞ�ðu1; . . . ; um�nÞ¼ ½WRða�nÞ�ðe;bÞðv�1; . . . ; v�nÞ ¼ 1:

Therefore, by virtue of Lemma 3.2(i), we observe that

1 ¼ ð½WRðamÞ�ðu1; . . . ; umÞÞ2 � ½WRða�nÞ�ðe;bÞðv�1; . . . ; v�nÞ

6 ½WRðam�nÞ�ðu1; . . . ; um�nÞ � EðumDv�n; um�nÞ ¼ 1 � EðumDv�n; um�nÞ¼ EðumDv�n; um�nÞ;

i.e. umDv�n �E um�n,

i:e: ½am�Rf�iji¼1;...;m�1gD ½a�n�RfDiji¼1;...;n�1g

� �ðe;bÞ

� ��E½am�n�Rf�i ji¼1;...;m�n�1g:

In a similar fashion to (i), by using Proposition 2.5, the properties (ii) and (iii)

can be easily obtained from Lemma 3.2(ii) and (iii), respectively. h

Proof of Theorem 3.1. Taking Definition 2.4 into consideration, the property

(i) follows from [3, Theorem 4.4(i)], at once. To prove the remaining properties,

let us assume that (X ; ~�) forms an M-vague group w.r.t. P and E. (i) directly

yields (ii). Thus, it is sufficient to verify the properties (iii–v).

1510 M. Demirci / Information Sciences 176 (2006) 1488–1530

(iii) Let the condition (a) be satisfied. Due to Proposition 2.2(vi), ~� is com-

pletely transitive. Let us take fDiji ¼ 1; . . . ;m� 1g � ORDð~�Þ. From

Lemma 3.3(i), we have

E ½am�RfDi ji¼1;...;m�1gD ½a�n�RfDi ji¼1;...;n�1g

� �ðe;bÞ

; am�n½ �RfDiji¼1;...;m�n�1g

� �¼ 1:

Then

E ½am�n�RfDi ji¼1;...;m�n�1g; z� �¼ E ½am�n�RfDi ji¼1;...;m�n�1g; z

� �� E ½am�n�RfDi ji¼1;...;m�n�1g; ½am�RfDiji¼1;...;m�1gD ½a�n�RfDiji¼1;...;n�1g

� �ðe;bÞ

� �6 E ½am�RfDi ji¼1;...;m�1gD ½a�n�RfDi ji¼1;...;n�1g

� �ðe;bÞ

; z� �

and similarly

E ½am�RfDi ji¼1;...;m�1gD ½a�n�RfDi ji¼1;...;n�1g

� �ðe;bÞ

; z� �

6 E ½am�n�RfDi ji¼1;...;m�n�1g; z� �

so we get

E ½am�n�RfDi ji¼1;...;m�n�1g; z� �

¼ E ½am�RfDi ji¼1;...;m�1g;D ½a�n�RfDiji¼1;...;n�1g

� �ðe;bÞ

; z� �

:

Therefore, applying Proposition 2.9(iii), and considering the completetransitivity of ~�, we observe that

½PwRðamÞ�ðxÞ� ½PwRða�nÞ�ðe;bÞðyÞ� ½PwRðam�nÞ�ðzÞ

6E ½am�RfDi ji¼1;...;m�1g;x� �

�E ½a�n�RfDi ji¼1;...;n�1g

� �ðe;bÞ

;y� �

�E am�n½ �RfDiji¼1;...;m�n�1g;z� �

¼E ½am�RfDi ji¼1;...;m�1g;x� �

�E ½a�n�RfDi ji¼1;...;n�1g

� �ðe;bÞ

;y� �

�E ½am�RfDi ji¼1;...;m�1gD ½a�n�RfDi ji¼1;...;n�1g

� �ðe;bÞ

;z� �

¼ E ½am�RfDi ji¼1;...;m�1g;x� �

� ~� ½am�RfDiji¼1;...;m�1g; ½a�n�RfDiji¼1;...;n�1g

� �ðe:bÞ

;z� � �

�E a�n½ �RfDiji¼1;...;n�1g

� �ðe;bÞ

;y� �

6 ~� x; ½a�n�RfDi ji¼1;...;n�1g

� �ðe;bÞ

;z� �

�E ½a�n�RfDiji¼1;...;n�1g

� �ðe;bÞ

;y� �

6 ~�ðx;y;zÞ6EðxDy;zÞ:

M. Demirci / Information Sciences 176 (2006) 1488–1530 1511

If the condition (b) is satisfied, then (X,vag(D)) forms a perfect M-vague

group, and vag(D) is obviously transitive of the first and third order.

Thus, we have

½PwRvagðDÞðamÞ�ðxÞ � ½PwR

vagðDÞða�nÞ�ðe;bÞðyÞ � ½PwRvagðDÞðam�nÞ�ðzÞ 6 EðxDy; zÞ:

ð3:1ÞBecause of the fact that

PwRðamÞ 6 PwRvagðDÞðamÞ; ½PwRða�nÞ�ðe;bÞ 6 ½PwR

vagðDÞða�nÞ�ðe;bÞand PwRðam�nÞ 6 PwR

vagðDÞðam�nÞ;

the required inequality follows from (3.1), at once. We will prove the

properties (iv) and (v) for assumption (a) only. Their proof for assump-

tion (b) is similar to (iii), and is skipped here.

(iv) For fDiji ¼ 1; . . . ; n� 1g � ORDð~�Þ, we have from Lemma 3.3(ii) that

E ½an�RfDi ji¼1;...;n�1gD ½a�n�RfDi ji¼1;...;n�1g

� �ðe;bÞ

; e� �

¼ 1:

Then, using Proposition 2.9(iii), and by virtue of the complete transitivity

of ~�, we see that

½PwRðanÞ�ðxÞ � ½PwRða�nÞ�ðe;bÞðyÞ

6 E ½an�RfDi ji¼1;...;n�1g; x� �

� E ½a�n�RfDi ji¼1;...;n�1g

� �ðe;bÞ

; y� �

¼ E ½an�RfDiji¼1;...;n�1g; x� �

� E ½a�n�RfDiji¼1;...;n�1g

� �ðe;bÞ

; y� �

� E ½an�RfDi ji¼1;...;n�1g;D ½an�RfDiji¼1;...;n�1g

� �ðe;bÞ

; e� �

¼ E ½an�RfDi ji¼1;...;n�1g; x� �

� ~� ½an�RfDi ji¼1;...;n�1g;

�½a�n�RfDi ji¼1;...;n�1g

� �ðe;bÞ

; e��

� E ½a�n�RfDiji¼1;...;n�1g

� �ðe;bÞ

; y� �

6 ~� x; ½a�n�RfDi ji¼1;...;n�1g

� �ðe;bÞ

; e� �

� E ½a�n�RfDi ji¼1;...;n�1g

� �ðe;bÞ

; y� �

6 ~�ðx; y; eÞ 6 EðxDy; eÞ:

(v) Let us pick fDiji ¼ 1; . . . ; n� 1g � ORDð~�Þ. In a similar fashion to (iii),by using Lemma 3.3(iii), Proposition 2.9(iii) and the complete transitivity

of ~�, we observe that

1512 M. Demirci / Information Sciences 176 (2006) 1488–1530

½PwRðamÞ�ðxÞ � ½PwRða�nÞ�ðe;bÞðyÞ � ½PwRða�ðn�mÞ�ðe;bÞðzÞ

6 E ½am�RfDiji¼1;...;m�1g; x� �

� E ½a�n�RfDiji¼1;...;n�1g

� �ðe;bÞ

; y� �

� E ½a�ðn�mÞ�RfDi ji¼1;...;n�m�1g

� �ðe;bÞ

; z� �

¼ E ½am�RfDiji¼1;...;m�1g; x� �

� E ½a�n�RfDiji¼1;...;n�1g

� �ðe;bÞ

; y� �

� E ½am�RfDiji¼1;...;m�1gD ½a�n�RfDiji¼1;...;n�1g

� �ðe;bÞ

; z� �

¼

E ½am�RfDiji¼1;...;m�1g; x� �

� ~� ½am�RfDi ji¼1;...;m�1g;�

½a�n�RfDiji¼1;...;n�1g

� �ðe;bÞ

; z��

� E ½a�n�RfDiji¼1;...;n�1g

� �ðe;bÞ

; y� �

6 ~� x; ½a�n�RfDiji¼1;...;n�1g

� �ðe;bÞ

; z� �

� E ½a�n�RfDi ji¼1;...;n�1g

� �ðe;bÞ

; y� �

6 ~�ðx; y; zÞ 6 EðxDy; zÞ: �

If we take Proposition 2.2(vi) into consideration, and if the condition (a) is

replaced by the first order transitivity of ~�, then it is easy to note that the prop-

erties (ii–v) in Theorem 3.1 still remain valid. Keeping in mind this observation,

and considering Proposition 2.9(iv) we easily observe that if the M-vague

semigroup ðX ; ~�Þ in the hypothesis of Theorem 3.1 is enriched to be a perfect

M-vague semigroup having an associative ordinary description of ~�, then all

properties in Theorem 3.1 can also be stated for vague integral powers insteadof right vague integral powers. This observation is formally introduced in the

following corollary.

Corollary 3.4. For an integral, commutative cl-monoid M = (L,6,*), let ðX ; ~�Þbe a perfect M-vague semigroup w.r.t. P and E, and D 2 ORDð~�Þ an associative

ordinary description of ~�.

(i) If ~� is transitive of the third order, then for m; n 2 Nþ and for alla,x,y, z 2 X,

½PwðanÞ�ðxÞ � ½PwðamÞ�ðyÞ � ½PwðanþmÞ�ðzÞ 6 EðxDy; zÞ:Furthermore, if ðX ; ~�Þ is an M-vague group w.r.t. P and E, then for

m; n 2 Nþ and for all a,x,y, z 2 X, e 2 Id�ðX ; ~�Þ; b 2 InvðX ;~�Þða; eÞ, the fol-

lowing properties are true:

(ii) ½Pwða�nÞ�ðe;bÞðxÞ � ½Pwða�mÞ�ðe;bÞðyÞ � ½Pwða�ðnþmÞÞ�ðe;bÞðzÞ 6 EðxDy; zÞ:

M. Demirci / Information Sciences 176 (2006) 1488–1530 1513

(iii) For m>n,

½PwðamÞ�ðxÞ � ½Pwða�nÞ�ðe;bÞðyÞ � ½Pwðam�nÞ�ðzÞ 6 EðxDy; zÞ:

(iv) For m = n

½PwðanÞ�ðxÞ � ½Pwða�nÞ�ðe;bÞðyÞ 6 EðxDy; eÞ:

(v) For m<n

½PwðamÞ�ðxÞ � ½Pwða�nÞ�ðe;bÞðyÞ � ½Pwða�ðn�mÞÞ�ðe;bÞðzÞ 6 EðxDy; zÞ:

Given a semigroup (group) (X,�) in the classical sense, for m, n 2 Nþðm; n 2 ZÞand a 2 X, the equality (am)n = amÆn is another well-known property of integral

powers of elements. In the remaining part of this section, we will study the

vague counterpart to this property. Our main aim is to formulate this property

for vague integral powers of elements. For this purpose, we first need to estab-

lish this property for graded successive integral powers of elements. The formu-

lation of vague counterpart to the considered property for graded successiveintegral powers is given in the subsequent lemmas.

Lemma 3.5. Let ðX ; ~�Þ be an M-vague semigroup w.r.t. P and E, and

D 2 ORDð~�Þ. Suppose that the statement ‘‘~� is transitive of the first and third

order, or E is regular w.r.t D’’ is true. Then for m; n 2 Nþ and for all

a,u1, . . . , umÆn, v1, . . . , vn 2 X, the following inequality is satisfied:

½WRðam�nÞ�ðu1; . . . ; um�nÞ� �n�1 � WRððumÞnÞ

� �ðv1; . . . ; vnÞ

� ½WRðamÞ�ðu1; . . . ; umÞ� �n�1

6 Eðum�n; vnÞ:

Remark 3.6. If ðX ; ~�Þ is taken as an M-vague group w.r.t. P and E in Lemma

3.5, then by Proposition 2.2(vi), it is easy to note that the sentence ‘‘~� is tran-

sitive of the first and third order’’ in the hypothesis of Lemma 3.5 can be simply

substituted by the first order transitivity of ~�.Lemma 3.7. Let ðX ; ~�Þ be an M-vague group w.r.t. P and E, and D 2 ORDð~�Þ.Assume that one of the conditions (a) or (b) in Lemma 3.2 is satisfied. Then for

n 2 Nþ and for all a; u1; . . . ; un; u�1; . . . ; u�n 2 X ; e; e0 2 Id�ðX ; ~�Þ; b 2 InvðX ;~�Þða; eÞ and c 2 InvðX ;~�Þðun; e0Þ,

½WRðanÞ�ðu1; . . . ; unÞ � ½WRða�nÞ�ðe;bÞðu�1; . . . ; u�nÞ 6 Eðc; u�nÞ:

Lemma 3.8. Given an M-vague group ðX ; ~�Þ w.r.t. P and E and D 2 ORDð~�Þ, let

one of the conditions (a) or (b) in Lemma 3.2 be fulfilled. Then for m; n 2 Nþ and

for all a; u1; . . . ; um�n; u�1; . . . ; u�m�n; v�1; . . . ; vn;2 X ; e; e0 2 Id�ðX ; ~�Þ; b 2InvðX ;~�Þða; eÞ; c 2 InvðX ;~�Þðum; e0Þ and d 2 InvðX ;~�Þðu�m; e0Þ, the following relations hold:

1514 M. Demirci / Information Sciences 176 (2006) 1488–1530

(i) ½WRða�m�nÞ�ðe;bÞðu�1; . . . ; u�m�nÞ� �n�1

� ½WRðamÞ�ðu1; . . . ; umÞ� �nþ1 � ½WR

�ða�mÞ�ðe;bÞðu�1; . . . ; u�mÞÞ2n � ½WRððumÞ�nÞ�ðe0 ;cÞðv�1; . . . ; v�nÞ 6 Eðu�m�n; v�nÞ:

(ii) ½WRðam�nÞ�ðu1; . . . ; um�nÞ� �n�1 � ½WRða�mÞ�ðe;bÞðu�1; . . . ; u�mÞ

� �nþ1

�½WRðamÞ�ðu1; . . . ; umÞ� �2n �½WRððu�mÞ�nÞ�ðe0 ;dÞðv�1; . . . ; v�nÞ 6 Eðum�n; v�nÞ:

The vague counterpart to the equality ‘‘(am)n = amÆn’’ for vague integral pow-

ers also requires the formulation of this equality by means of exact integralpowers. The required formulation is given in the following two propositions.

Proposition 3.9. Let ðX ; ~�Þ be an M-vague semigroup w.r.t. P and E. Under the

hypothesis of Lemma 3.5, for m; n 2 Nþ, a 2 X and {Diji = 1, . . . ,m Æn � 1},

f�iji ¼ 1; . . . ; n� 1g � ORDð~�Þ we have

½am�n�RfDiji¼1;...;m�n�1g�E ½am�RfDiji¼1;...;m�1g

� �nh iRf�i ji¼1;...;n�1g:

Proof. Particularly, choosing the elements u1,u2, . . . ,umÆn,v1, . . . ,vn 2 X in

Lemma 3.5 such that

uk ¼ ½ak�RfDi ji¼1;...;k�1g and vl ¼ ½am�RfDiji¼1;...;m�1g

� �l �R

f�i ji¼1;...;l�1g

for all k = 1, . . . ,mÆn and l = 1, . . . ,n, the required relation follows from Propo-

sition 2.5(i) and Lemma 3.5. h

Proposition 3.10. Let ðX ; ~�Þ be an M-vague group w.r.t. P and E provided that

the conditions (a) or (b) in Lemma 3.2 is satisfied. For m; n 2 Nþ;a 2 X ;e;e0 2 Id�ðX ; ~�Þ and f�iji¼ 1; . . . ;m� 1g � ORDð~�Þ, let b 2 InvðX ;~�Þ ða;eÞ,

c 2 InvðX ;~�Þ ½am�Rf�i ji¼1;...;m�1g;e0

� �and d 2 InvðX ;~�Þ ½a�m�Rf�i ji¼1;...;m�1g

� �ðe;bÞ

;

�e0Þ. Then

for all {Diji = 1, . . . ,m Æn � 1}, fuiji¼ 1; . . . ;n� 1g � ORDð~�Þ, the following rela-

tions are valid:

(i) ½a�m�n�RfDi ji¼1;...;m�n�1g

� �ðe;bÞ�E ½am�Rf�iji¼1;...;m�1g

� ��nh iRfuiji¼1;...;n�1g

� �ðe0 ;cÞ

:

(ii) ½am�n�RfDiji¼1;...;m�n�1g�E ½a�m�Rf�i ji¼1;...;m�1g

� �ðe;bÞ

� ��n �Rfuiji¼1;...;n�1g

!ðe0 ;dÞ

:

Proof. Taking Proposition 2.5(i) and (ii) into consideration, and employing the

same technique in Proposition 3.9, the properties (i) and (ii) can be easily

deduced from Lemma 3.8(i) and (ii), respectively. h

M. Demirci / Information Sciences 176 (2006) 1488–1530 1515

In the remaining part of the paper, we assume that M = (L,6, *) is an inte-

gral, commutative cl-monoid. Following these preparatory results, we are now

in a position to formulate the equality (am)n = amÆn for vague integral powers of

elements in vague groups. We first give this formulation for the case m; n 2 Nþ:

Theorem 3.11. For a given M-vague semigroup ðX ; ~�Þ w.r.t. P and E and forD 2 ORDð~�Þ, let us consider the conditions:

(a) ~� is transitive of the first and third order,

(b) ~� is completely transitive,

(c) E is regular w.r.t. D.

Then for m; n 2 Nþ and for all, a; x; y; z 2 X ; fDiji ¼ 1; . . . ;m� 1g � ORDð~�Þthe following statements are true:

(i) If the conditions (a) or (c) are satisfied, then

½PwRðam�nÞ�ðxÞ � PwR ½am�RfDiji¼1;...;m�1g

� �n� �h iðyÞ 6 Eðx; yÞ:

(ii) If the conditions (b) or (c) are fulfilled, then

½PwRðam�nÞ�ðxÞ � ½PwRðamÞ�ðzÞ� �n � ½PwRðznÞ�ðyÞ 6 Eðx; yÞ:

Proof. (i) Suppose that (a) or (c) is satisfied. Let us pick {Diji = 1, . . . ,mÆn � 1},

f�iji ¼ 1; . . . ; n� 1g � ORDð~�Þ. From Proposition 3.9, we have

E ½am�n�RfDiji¼1;...;m�n�1g; ½am�RfDiji¼1;...;m�1g

� �nh iRf�i ji¼1;...;n�1g

� �¼ 1:

Therefore, by using Proposition 2.9(iii), it is easy to see that

½PwRðam�nÞ�ðxÞ � PwR ½am�RfDiji¼1;...;m�1g

� �n� �h iðyÞ

6 E ½am�n�RfDiji¼1;...;m�n�1g; x� �

� E ½am�RfDiji¼1;...;m�1g

� �nh iRf�iji¼1;...;n�1g

; y� �

¼ E ½am�n�RfDi ji¼1;...;m�n�1g; x� �

� E ½am�RfDiji¼1;...;m�1g

� �nh iRf�i ji¼1;...;n�1g

; y� �

� E ½am�n�RfDi ji¼1;...;m�n�1g; ½am�RfDi ji¼1;...;m�1g

� �nh iRf�i ji¼1;...;n�1g

� �6 Eðx; yÞ:

(ii) Assume that (b) or (c) is true. In an analogous way to Theorem 3.1, it is

sufficient to prove the assertion for the case (b). Using the property (i), and

considering Proposition 2.9(iii) and Proposition 2.7(i), one can easily observe

that

1516 M. Demirci / Information Sciences 176 (2006) 1488–1530

½PwRðam�nÞ�ðxÞ � PwRðamÞ� �

ðzÞ� �n � PwRðznÞ

� �ðyÞ

6 ½PwRðam�nÞ�ðxÞ � E ½am�RfDiji¼1;...;m�1g; z� �� �n

� PwRðznÞ� �

ðyÞ

6 ½PwRðam�nÞ�ðxÞ � PwR ½am�RfDiji¼1;...;m�1g

� �n� �h iðyÞ 6 Eðx; yÞ: �

The vague counterpart to the equality with (am)n = amÆn for m; n 2 Z� f0gwith ðm; nÞ 62 Nþ Nþ is formulated by means of vague integral powers of ele-

ments in the theorem given below.

Theorem 3.12. Let ðX ; ~�Þ be an M-vague group w.r.t. P and E provided that one

of the conditions (a) or (b) in Lemma 3.2 is satisfied. Then for m; n 2 Nþ and for

all a; x; y; z 2 X ; e; e0 2 Id�ðX ; ~�Þ; b 2 InvðX ;~�Þða; eÞ; v 2 InvðX ;~�Þðx; e0Þ and

w 2 InvðX ;~�Þðz; e0Þ the following properties are valid:

(i) PwRðanÞ� �

ðxÞ � PwRða�nÞ� �

ðe;bÞðyÞ 6 Eðv; yÞ:(ii) PwRða�m�nÞ

� �ðe;bÞðxÞ � ½PwRðamÞ�ðzÞ

� �n � PwRðz�nÞ� �

ðe0 ;wÞðyÞ 6 Eðx; yÞ:(iii) PwRðam�nÞ

� �ðxÞ � ½PwRða�mÞ�ðe;bÞðzÞ

� �n� PwRðz�nÞ� �

ðe0;wÞðyÞ 6 Eðx; yÞ:

Proof. (i) Since the required inequality is clear for n = 1, it is enough to verify

the assertion for n P 2. For {�iji = 1, . . . ,n � 1}, fDiji ¼ 1; . . . ; n� 1g �ORDð~�Þ, let us particularly choose the elements c,u1, . . . ,un,u�1, . . . ,u�n 2 X

in Lemma 3.7 such that un = x, u�n = y c = v, uk ¼ ½ak�Rf�iji¼1;...;k�1g and

u�k ¼ ½a�k�RfDi ji¼1;...;k�1g

� �ðe;bÞ

for all k = 1, . . . ,n � 1. Then considering the equal-

ity (2.4), we obtain from Lemma 3.7 that

~� ½an�1�Rf�i ji¼1;...;n�2g; a; x� �

� ~� ½a�ðn�1Þ�RfDi ji¼1;...;n�2g

� �ðe;bÞ

; b; y� �

6 Eðv; yÞ:

Then taking the supremum over {�iji = 1, . . . ,n � 2} and {Diji = 1, . . . ,n � 2} atthe left-hand side of this inequality, and using the distributivity of * over arbi-

trary joins, we observe that

PwRðanÞ� �

ðxÞ � PwRða�nÞ� �

ðe;bÞðyÞ

¼_ ~� ½an�1�Rf�i ji¼1;...;n�2g;a;x� �

jf�ij ¼ 1; . . . ;n�2g�ORDð~�Þn o

�_ ~� ½a�ðn�1Þ�RfDiji¼1;...;n�2g

� �ðe;bÞ

;b;y�� ����fDiji¼ 1; . . . ;n�2g�ORDð~�Þ

� ¼_ ~� ½an�1�Rf�iji¼1;...;n�2g;a;x

� �� ~� ½a�ðn�1Þ�RfDi ji¼1;...;n�2g

� �ðe;bÞ

;b;y�� �����

f�iji¼ 1; ; . . . ;n�2g;fDiji¼ 1; . . . ;n�2g�ORDð~�Þ6Eðv;yÞ:

M. Demirci / Information Sciences 176 (2006) 1488–1530 1517

Before introducing proofs of the properties (ii) and (iii), for an arbitrarily fixed

{Diji = 1, . . . ,m Æn � 1}, fuiji ¼ 1; . . . ; n� 1g � ORDð~�Þ, let us first take

c 2 InvðX ;~�Þ ½am�RfDiji¼1;...;m�1g; e0

� �and d 2 InvðX ;~�Þ ½a�m�RfDiji¼1;...;m�1g

� �ðe;bÞ

; e0� �

.

(ii) It is clear from Proposition 2.2(viii) that

E ½am�RDi ji¼1;...;m�1; z� �

¼ Eðc;wÞ:

Furthermore, by Proposition 3.10(i), we have

E ½a�m�n�RfDiji¼1;...;m�n�1g

� �ðe;bÞ

; ½am�RfDi ji¼1;...;m�1g

� ��nh iRfui ji¼1;...;n�1g

� �ðe0;cÞ

!¼ 1:

Therefore, using Proposition 2.9(iii) and Proposition 2.7(i), we observe that

½PwRða�m�nÞ�ðe;bÞðxÞ� PwRðamÞ� �

ðzÞ� �n � PwRðz�nÞ

� �ðe0 ;wÞðyÞ

6 PwRða�m�nÞ� �

ðe;bÞðxÞ� E ½am�RDi ji¼1;...;m�1;z� �� �n

� PwRðwnÞ� �

ðyÞ

¼ PwRða�m�nÞ� �

ðe;bÞðxÞ�ðEðc;wÞÞn � PwRðwnÞ� �

ðyÞ

6 PwRða�m�nÞ� �

ðe;bÞðxÞ� PwRðcnÞ� �

ðyÞ

¼ PwRða�m�nÞ� �

ðe;bÞðxÞ� PwR ½am�RfDiji¼1;...;m�1g

� ��n� �h iðe0 ;cÞðyÞ

6E ½a�m�n�RfDi ji¼1;...;m�n�1g

� �ðe;bÞ

;x� ��E ½am�RfDi ji¼1;...;m�1g

� ��nh iRfui ji¼1;...;n�1g

� �ðe0;cÞ

;y

!

¼E ½a�m�n�RfDi ji¼1;...;m�n�1g

� �ðe;bÞ

;x� ��E ½am�RfDiji¼1;...;m�1g

� ��nh iRfui ji¼1;...;n�1g

� �ðe0;cÞ

;y

!

�E ½a�m�n�RfDi ji¼1;...;m�n�1g

� �ðe;bÞ

; ½am�RfDi ji¼1;...;m�1g

� ��nh iRfuiji¼1;...;n�1g

� �ðe0 ;cÞ

!6Eðx;yÞ:

(iii) Similar to (ii), Proposition 2.2(viii) gives that

E ½a�m�RfDi ji¼1;...;m�1g

� �ðe;bÞ

; z� �

¼ Eðd;wÞ:

1518 M. Demirci / Information Sciences 176 (2006) 1488–1530

From Proposition 3.10(ii), we have

E ½a�m�n�RfDi ji¼1;...;m�n�1g

� �ðe;bÞ

;

�½a�m�RfDi ji¼1;...;m�1g

� �ðe;bÞ

� ��n �Rfui ji¼1;...;n�1g

!ðe0 ;dÞ

1A¼ 1:

Thus, by virtue of Propositions 2.9(iii) and 2.7(i), we can write

½PwRðam�nÞ�ðxÞ� ½PwRða�mÞ�ðe;bÞðzÞ� �n

� ½PwRðz�nÞ�ðe0 ;wÞðyÞ

6 ½PwRðam�nÞ�ðxÞ � E ð½a�m�RfDi ji¼1;...;m�1gÞðe;bÞ;z� �h in

� ½PwRðwnÞ�ðyÞ

¼ ½PwRðam�nÞ�ðxÞ � ðEðd;wÞÞn � ½PwRðwnÞ�ðyÞ6 ½PwRðam�nÞ�ðxÞ � ½PwRðdnÞ�ðyÞ

¼ ½PwRðam�nÞ�ðxÞ � PwR ð½a�m�RfDi ji¼1;...;m�1gÞðe;bÞ� ��n� �h i

ðe0;dÞðyÞ

6E ð½a�m�n�RfDi ji¼1;...;m�n�1gÞðe;bÞ;x� ��E ðð½a�m�RfDiji¼1;...;m�1gÞðe;bÞÞ

�nh iR

fuiji¼1;...;n�1g

� �ðe0;dÞ

;y

!¼E ð½a�m�n�RfDiji¼1;...;m�n�1gÞðe;bÞ;x

� ��E ½ðð½a�m�RfDiji¼1;...;m�1gÞðe;bÞÞ

�n�Rfuiji¼1;...;n�1g

� �ðe0 ;dÞ

;y� �

�E ½a�m�n�RfDiji¼1;...;m�n�1g

� �ðe;bÞ

;ð½ðð½a�m�RfDiji¼1;...;m�1gÞðe;bÞÞ�n�Rfuiji¼1;...;n�1gÞðe0;dÞ

� �6Eðx;yÞ: �

In a similar fashion to Corollary 3.4, if the M-vague semigroup (group)

ðX ; ~�Þ in Theorem 3.11 (Theorem 3.12) is strengthened to be a perfect M-vague

semigroup (group) an associative ordinary description of ~�, then Theorems

3.11 and 3.12 can also be expressed for vague integral powers instead of right

vague integral powers. This can be formally elucidated in the following

corollary.

Corollary 3.13. Given a perfect M-vague semigroup ðX ; ~�Þ w.r.t. P and E, let

D 2 ORDð~�Þ be an associative ordinary description of ~�.

(i) If ~� is transitive of the third order, or equivalently E is right regular w.r.t. D,

then for m; n 2 Nþ and for all a,x,y 2 X,

½Pwðam�nÞ�ðxÞ � ½Pwðð½am�DÞnÞ�ðyÞ 6 Eðx; yÞ:

(ii) If ~� is completely transitive, or equivalently E is regular w.r.t. D, then for

m; n 2 Nþ and for all a,x,y, z 2 X,

M. Demirci / Information Sciences 176 (2006) 1488–1530 1519

½Pwðam�nÞ�ðxÞ � ð½PwðamÞ�ðzÞÞn � ½PwðznÞ�ðyÞ 6 Eðx; yÞ:Furthermore, if ðX ; ~�Þ is an M-vague group w.r.t. P and E, then for m, n 2 Nþ

and for all a; x; y; z 2 X ; e; e0 2 Id�ðX ; ~�Þ; b 2 InvðX ;~�Þða; eÞ; v 2 InvðX ;~�Þðx; e0Þand w 2 InvðX ;~�Þðz; e0Þ, the following properties are also valid:

(iii) ½PwðanÞ�ðxÞ � ½Pwða�nÞ�ðe;bÞðyÞ 6 Eðv; yÞ.

(iv) ½Pwða�m�nÞ�ðe;bÞðxÞ � ð½PwðamÞ�ðzÞÞn � ½Pwðz�nÞ�ðe0 ;wÞðyÞ 6 Eðx; yÞ:

½Pwðam�nÞ�ðxÞ � ð½Pwða�mÞ� ðzÞÞn � ½Pwðz�nÞ� ðyÞ 6 Eðx; yÞ:

(v) ðe;bÞ ðe0 ;wÞ

4. Conclusion

In classical mathematics, the real numbers are assumed to be characterized

and manipulated with infinite precision. However this assumption does not en-

able us to handle the uncertainty occurring in a measurement process of a

quantity q. Here we practically do not obtain a certain real number as the ac-

tual value of q, but some data points close to a typical value x. This means that

the actual value of q obtained in a practical measurement process will not be areal number with infinite precision, but the set ~x of data points close to x. ~x can

be thought of as the fuzzy set of real numbers approximately equal to x, and ~xcan be simply modelled as the extensional hull [x]E of x according to some

suitably chosen M-equivalence relation E. If we vary the amount of the quan-

tity q by n-times, then we practically obtain the fuzzy set exk of real numbers

approximately equal to xk corresponding to the actual value of q for

k = 1,2, . . . ,n. It is intuitively natural to expect that the sum (product) ofex1 ; ex2 ; . . . ; exn should be the fuzzy set gPnk¼1xk

gðQnk¼1xkÞ of real numbers approx-

imately equal toPn

k¼1xk ðQn

k¼1xkÞ. If we particularly consider ~þ ð~�Þ as a perfect

M-vague addition (multiplication) operation on R, and define the the

sum (product) of ex1 ; ex2 ; . . . ; exn as the vague sum fPðx1; x2; . . . ; xnÞ (the vague

product fQðx1; x2; . . . ; xnÞÞ of x1,x2, . . . ,xn w.r.t ~þ ð~�Þ, then this intuitive expec-

tation is satisfied by fPðx1; x2; . . . ; xnÞ ðfQðx1; x2; . . . ; xnÞÞ (see [3, Proposition

3.6(iii)]), i.e.

gXðx1; x2; . . . ; xnÞ ¼gXn

k¼1

xkfYðx1; x2; . . . ; xnÞ ¼

gYn

k¼1

xk

!:

Similarly, the mth multiple (power) of an actual value ~x of q is naturally antic-

ipated to be the fuzzy set gm:x ðfxmÞ of real numbers approximately equal to mÆx(xm). In this case, if we define the mth multiple (power) of ~x as the vague mth

1520 M. Demirci / Information Sciences 176 (2006) 1488–1530

multiple Pw(mx) (the vague mth power Pw(xm)) of x w.r.t. ~þ ð~�Þ, then the

formation of Pw(mx) (Pw(xm)) fits this anticipation (see Proposition 2.9(iv)),

i.e.

PwðmxÞ ¼ gm � x ðPwðxmÞ ¼ fxmÞ:

As explained in the above paragraphs, the vague sum (product) of a finite num-

ber of real numbers best suits the mathematical modelling of the sum (product)

of a finite number of the fuzzy real numbers approximately equal to some real

numbers. The same situation is also valid for the vague integral multiplies

(powers) of real numbers. From the view-point of information sciences, the

underlying M-equivalence relation E carries the information of indistinguish-

ability of any two real numbers. This information results in fuzzy numbers

approximately equal to real numbers instead of the real numbers with infiniteprecision, and the notions of vague sum, vague product, vague integral multi-

ple and power of real numbers become natural formulation of the sum, prod-

uct, integral multiple and power of fuzzy real numbers approximately equal to

some real numbers. Various fundamental properties of these notions basically

depend on a many-valued counterpart to the generalized associative law (called

the generalized vague associative law [3]). For this reason, a considerable part

of this paper together with Part I has been devoted to this subject, and it has

been demonstrated in the present paper how some elementary properties of thevague integral multiples and powers can be established by making use of the

generalized vague associative law.

The notion of integral power of elements, which is nothing but a special case

of the vague integral power of elements in the two-valued logic, forms an essen-

tial tool of classical group theory, and it plays a major role in various branches

of classical algebra and its applications, e.g. finite generated groups, cyclic

groups, free groups, free abelian groups [5], etc. This paper paves the way

for the development of vague algebraic structures involving vague integralpowers of elements. For instance, vague integral powers of elements could

be applied to vague counterparts to finitely generated groups, cyclic groups,

free groups, free abelian groups. In this paper, we restrict our attention to only

the introduction of this notion, and investigate some of its fundamental

properties.

Appendix A. Proof of Lemma 3.2

For u1 5 a or v�1 5 b, all properties are trivial. Let us assume that u1 = a

and v�1 = b.

(i) Suppose that m > n, and the condition (a) is satisfied. Let us choose

um+1,um+2, . . . ,um+n 2 X such that

M. Demirci / Information Sciences 176 (2006) 1488–1530 1521

~�ðum; b; umþ1Þ ¼ ~�ðumþ1; b; umþ2Þ ¼ � � � ¼ ~�ðumþn�1; b; umþnÞ ¼ 1:

Then, since

WR ða; . . . ; a|fflfflfflffl{zfflfflfflffl}m times

; b; b; . . . ; b|fflfflfflfflfflffl{zfflfflfflfflfflffl}n times

Þ; ðu1; . . . ; um; umþ1; . . . ; umþnÞ

24 35¼ ½WRðamÞ�ðu1; . . . ; umÞ;

and by virtue of [3, Lemma 4.1(i)], we can write

½WRðamÞ�ðu1; . . . ; umÞ � ½WRða�nÞ�ðe;bÞðv�1; . . . ; v�nÞ

¼ WR ða; . . . ; a|fflfflfflffl{zfflfflfflffl}m times

; b; b; . . . ; b|fflfflfflfflfflffl{zfflfflfflfflfflffl}n times

Þ; ðu1; . . . ; umþ1; . . . ; umþnÞ

24 35� ½WRða�nÞ�ðe;bÞðv�1; . . . ; v�nÞ

6 EðumDv�n; umþnÞ: ðA:1Þ

Recalling the equality (2.2), we have

~�ðum�1; a; umÞ ¼ ~�ðum; b; um�1Þ � ~�ðum; b; umþ1Þ6 EðumDb; um�1Þ � EðumDb; umþ1Þ 6 Eðum�1; umþ1Þ:

From the first-order transitivity of ~�, and using (2.2) again, we can write

~�ðum�2; a; um�1Þ � Eðum�1; umþ1Þ 6 ~�ðum�2; a; umþ1Þ ¼ ~�ðumþ1; b; um�2Þ:Therefore, we get

~�ðum�2; a; um�1Þ � ~�ðum�1; a; umÞ6 ~�ðum�2; a; um�1Þ � Eðum�1; umþ1Þ 6 ~�ðumþ1; b; um�2Þ¼ ~�ðumþ1; b; um�2Þ � ~�ðumþ1; b; umþ2Þ6 Eðumþ1Db; um�2Þ � Eðumþ1Db; umþ2Þ 6 Eðum�2; umþ2Þ:

Pursuing this procedure, we reach the inequality

~�ðum�n; a; um�nþ1Þ � � � � � ~�ðum�2; a; um�1Þ � ~�ðum�1; a; umÞ½ �6 Eðum�n; umþnÞ: ðA:2Þ

Now multiplying the inequalities (A.1) and (A.2) side by side w.r.t. *, we get

½WRðamÞ�ðu1; . . . ; umÞ � ½WRða�nÞ�ðe;bÞðv�1; . . . ; v�nÞ � ½~�ðum�n; a; um�nþ1Þ

� � � � � ~�ðum�2; a; um�1Þ � ~�ðum�1; a; umÞ� 6 EðumDv�n; umþnÞ � Eðum�n; umþnÞ6 EðumDv�n; um�nÞ: ðA:3Þ

1522 M. Demirci / Information Sciences 176 (2006) 1488–1530

Then, considering the fact that

½WRðamÞ�ðu1; . . . ; umÞ ¼ ½WRðam�nÞ�ðu1; . . . ; um�nÞ � ½~�ðum�n; a; um�nþ1Þ � � � �� ~�ðum�2; a; um�1Þ � ~�ðum�1; a; umÞ�;

and multiplying both sides of (A.3) by ½WRðam�nÞ�ðu1; . . . ; um�nÞ w.r.t *, we see

that

ð½WRðamÞ�ðu1; . . . ; umÞÞ2 � ½WRða�nÞ�ðe;bÞðv�1; . . . ; v�nÞ

¼ ½WRðam�nÞ�ðu1; . . . ; um�nÞ � ð½WRðamÞ�ðu1; . . . ; umÞ� ½WRða�nÞ�ðe;bÞðv�1; . . . ; v�nÞ � ½~�ðum�n; a; um�nþ1Þ � � � �

� ~�ðum�2; a; um�1Þ � ~�ðum�1; a; umÞ�Þ6 WRðam�nÞ�ðu1; . . . ; um�nÞ � EðumDv�n; um�nÞ:

Let us now give the proof of the required inequality under the assumption(b). The proof is similar to [3, Lemma 4.1(ii)], and it is sufficient to show the

inequality (A.3) under the assumption (b). Due to the regularity of E w.r.t.

D, we obtain from Proposition 2.2(v) that (X, vag(D)) is a perfect M-vague

group w.r.t. P and E. Therefore, since vag(D) is a perfect M-vague binary oper-

ation w.r.t. P and E, the inequality (A.3) can be written for vag(D) instead of ~�,i.e.

½WRvagðDÞðamÞ�ðu1; . . . ; umÞ � ½WR

vagðDÞða�nÞ�ðe;bÞðv�1; . . . ; v�nÞ � ½vagðDÞ

ðum�n; a; um�n�1Þ � � � � � vagðDÞðum�2; a; um�1Þ � vagðDÞðum�1; a; umÞ�6 EðumDv�n; um�nÞ: ðA:4Þ

Hence the inequality (A.3) follows from the inequality (A.4) and the facts that~� 6 vagðDÞ; ½WRðamÞ� 6 ½WR

vagðDÞðamÞ� and ½WRða�nÞ� 6 ½WRvagðDÞða�nÞ�.

Now we only give the proof of the properties (ii) and (iii) under the assump-

tion (a). The proof for the case (b) can be easily done in a similar fashion to (i),so it is omitted here.

(ii) Assume that m = n. If n = 1, we obviously have

½WRðanÞ�ðu1; . . . ; unÞÞ2 � ½WRða�nÞ�ðe;bÞðv�1; . . . ; v�nÞ ¼ 1 ¼ ~�ða; b; eÞ¼ Eðu1Dv�1; eÞ;

so the required inequality is trivial. Consider the case n P 2. Let un+1,

un+2 . . . ,u2n 2 X be chosen such that

~�ðun; b; unþ1Þ ¼ ~�ðunþ1; b; unþ2Þ ¼ � � � ¼ ~�ðu2n�1; b; u2nÞ ¼ 1:

M. Demirci / Information Sciences 176 (2006) 1488–1530 1523

Then, since

WR ða; . . . ; a|fflfflfflffl{zfflfflfflffl}n times

b; . . . ; b|fflfflfflffl{zfflfflfflffl}n times

Þ; ðu1; . . . ; un; unþ1; . . . ; u2nÞ

24 35 ¼ ½WRðanÞ�ðu1; . . . ; unÞ;

we deduce from [3, Lemma 4.1(i)] that

½WRðanÞ�ðu1; . . . ; unÞ � ½WRða�nÞ�ðe;bÞðv�1; . . . ; v�nÞ

¼ WR ða; . . . ; a|fflfflfflffl{zfflfflfflffl}n times

; b; b; . . . ; b|fflfflfflfflfflffl{zfflfflfflfflfflffl}n times

Þ; ðu1; . . . ; un; unþ1; . . . ; u2nÞ

24 35� ½wRða�nÞ�ðe;bÞðv�1; . . . ; v�nÞ

6 EðunDv�n; u2nÞ: ðA:5Þ

Furthermore, the inequality (A.2) implies

½~�ðu1; a; u2Þ � � � � � ~�ðun�2; a; un�1Þ � ~�ðun�1; a; unÞ� 6 Eðu1; u2n�1Þ: ðA:6Þ

Then, using Proposition 2.2(vi) and the inequality (A.6), one can easily see that

½WRðanÞ�ðu1; . . . ; unÞ ¼ ½~�ðu1; a; u2Þ � . . . � ~�ðun�2; a; un�1Þ � ~�ðun�1; a; unÞ�6 Eðu1; u2n�1Þ ¼ Eðu1; u2n�1Þ � ~�ðu2n�1; b; u2nÞ6 ~�ðu1; b; u2nÞ ¼ ~�ða; b; u2nÞ:

Therefore, since

~�ða; b; u2nÞ ¼ ~�ða; b; u2nÞ � ~�ða; b; eÞ 6 EðaDb; u2nÞ � EðaDb; eÞ 6 Eðu2n; eÞ;

we get

½WRðanÞ�ðu1; . . . ; unÞ 6 ~�ða; b; u2nÞ 6 Eðu2n; eÞ: ðA:7ÞAt last, multiplying the inequalities (A.5) and (A.7) side by side w.r.t. *, we ob-

tain that

ð½WRðanÞ�ðu1; . . . ; unÞÞ2 � ½WRða�nÞ�ðe;bÞðv�1; . . . ; v�nÞ6 EðunDv�n; u2nÞ � Eðu2n; eÞ 6 EðunDv�n; eÞ.

(iii) Let m < n. In a similar fashion to (i), let us first consider the elementsum+1,um+2, . . .,um+n 2 X such that

~�ðum; b; umþ1Þ ¼ ~�ðumþ1; b; umþ2Þ ¼ . . . ¼ ~�ðumþn�1; b; umþnÞ ¼ 1:

The inequality (A.7) can be rewritten as

½WRðamÞ�ðu1; . . . ; umÞ 6 Eðu2m; eÞ:

1524 M. Demirci / Information Sciences 176 (2006) 1488–1530

Furthermore, considering Proposition 2.2(vi), we have

Eðu2m; eÞ ¼ Eðu2m; eÞ � ~�ðu2m; b; u2mþ1Þ 6 ~�ðe; b; u2mþ1Þ¼ ~�ðe; b; u2mþ1Þ � ~�ðe; b; bÞ 6 EðeDb; u2mþ1Þ � EðeDb; bÞ6 Eðb; u2mþ1Þ ¼ Eðb; u2mþ1Þ � ~�ðu2mþ1; b; u2mþ2Þ 6 ~�ðb; b; u2mþ2Þ:

Thus we get

½WRðamÞ�ðu1; . . . ; umÞ 6 ~�ðb; b; u2mþ2Þ: ðA:8ÞFor n � m P 2 and for all k = 1, . . . ,n � m � 1, taking Proposition 2.2(vi) into

consideration, we can write

Eðu2mþk; v�kÞ � ~�ðv�k; b; v�ðkþ1ÞÞ 6 ~�ðu2mþk; b; v�ðkþ1ÞÞ¼ ~�ðu2mþk; b; v�ðkþ1ÞÞ � ~�ðu2mþk; b; u2mþkþ1Þ 6 Eðu2mþkþ1; v�ðkþ1ÞÞ: ðA:9Þ

Now using the inequality (A.9), we observe that

~�ðb; b; u2mþ2Þ � ½WRða�ðn�mÞÞ�ðe;bÞðv�1; . . . ; v�ðn�mÞÞ¼ ~�ðb; b; u2mþ2Þ � ½~�ðb; b; v�2Þ � ~�ðv�2; b; v�3Þ � � � �� ~�ðv�ðn�m�1Þ; b; v�ðn�mÞ�¼ ½~�ðb; b; u2mþ2Þ � ~�ðb; b; v�2Þ� � ½~�ðv�2; b; v�3Þ � � � �� ~�ðv�ðn�m�1Þ; b; v�ðn�mÞ�6 Eðu2mþ2; v�2Þ � ½~�ðv�2; b; v�3Þ � � � � � ~�ðv�ðn�m�1Þ; b; v�ðn�mÞÞ�¼ ½Eðu2mþ2; v�2Þ � ~�ðv�2; b; v�3Þ� � ½~�ðv�3; b; v�4Þ � � � �� ~�ðv�ðn�m�1Þ; b; v�ðn�mÞ�6 Eðu2mþ3; v�3Þ � ½~�ðv�3; b; v�4Þ � � � � � ~�ðv�ðn�m�1Þ; b; v�ðn�mÞÞ�

..

.

6 Eðumþn�1; v�ðn�m�1ÞÞ � ~�ðv�ðn�m�1Þ; b; v�ðn�mÞÞ6 Eðumþn; v�ðn�mÞÞ:

Therefore, for n � m P 2, we find that

~�ðb; b; u2mþ2Þ � ½WRða�ðn�mÞÞ�ðe;bÞðv�1; . . . ; v�ðn�mÞÞ 6 Eðumþn; v�ðn�mÞÞ:ðA:10Þ

It is not difficult to see that the inequality (A.10) will also be valid for

n � m = 1. Thus, (A.10) is true for m < n. Then, we deduce from the inequali-

ties (A.8) and (A.10) that

½WRðamÞ�ðu1; . . . ; umÞ � ½WRða�ðn�mÞÞ�ðe;bÞðv�1; . . . ; v�ðn�mÞÞ

6 ~�ðb; b; u2mþ2Þ � ½WRða�ðn�mÞÞ�ðe;bÞðv�1; . . . ; v�ðn�mÞÞ6 Eðumþn; v�ðn�mÞÞ: ðA:11Þ

M. Demirci / Information Sciences 176 (2006) 1488–1530 1525

On the other hand, recalling [3, Lemma 4.1(i)], we have

½WRðamÞ�ðu1; . . . ; umÞ � ½WRða�nÞ�ðe;bÞðv�1; . . . ; v�nÞ

¼ WR ða; . . . ; a|fflfflfflffl{zfflfflfflffl};m times

b; b; . . . ; b|fflfflfflfflfflffl{zfflfflfflfflfflffl}n times

Þ; ðu1; . . . ; um; umþ1; . . . ; umþnÞ

24 35� ½WRða�nÞ�ðe;bÞðv�1; . . . ; v�nÞ 6 EðumDv�n; umþnÞ: ðA:12Þ

Finally, multiplying the inequalities (A.11) and (A.12) side by side w.r.t. *,

we obtain that

ð½WRðamÞ�ðu1; . . . ; umÞÞ2 � ½WRða�nÞ�ðe;bÞðv�1; . . . ; v�nÞ

� ½WRða�ðn�mÞ�ðe;bÞðv�1; . . . ; v�ðn�mÞÞ6 EðumDv�n; umþnÞ � Eðumþn; v�ðn�mÞ 6 EðumDv�n; v�ðn�mÞÞ: �

Appendix B. Proof of Lemma 3.5

Let ~� be transitive of the first and third order. Without loss of the generality,

we can assume that u1 = a and v1 = um. In order to verify the assertion, we ap-

ply induction on n. For n = 1, we have

ð½WRðam�nÞ�ðu1; . . . ; um�nÞÞn�1 � ½WRððumÞnÞ�ðv1; . . . ; vnÞ

� ð½WRðamÞ�ðu1; . . . ; umÞÞn�1 ¼ ½WRðumÞ�ðv1Þ ¼ 1 ¼ Eðum; v1Þ;

so the assertion is true for n = 1. Now assume that the assertion is true for

n � 1 (n P 2), i.e.

ð½WRðam�ðn�1ÞÞ�ðu1; . . . ; um�ðn�1ÞÞÞn�2 � ½WRððumÞn�1Þ�ðv1; . . . ; vn�1Þ

� ½WRðamÞ�ðu1; . . . ; umÞÞn�26 Eðum�ðn�1Þ; vn�1Þ: ðB:1Þ

[3, Lemma 4.1(i)] directly yields that

½WRðam�nÞ�ðu1; . . . ; um�nÞ � ½WRðamÞ�ðu1; . . . ; umÞ 6 Eðum�ðn�1ÞDum; um�nÞ:ðB:2Þ

On the other hand, since

½WRðam�nÞ�ðu1; . . . ; um�nÞ¼ ½WRðam�ðn�1ÞÞ�ðu1; . . . ; um�ðn�1ÞÞ � ~�ðum�ðn�1Þ; a; um�ðn�1Þþ1Þ� ~�ðum�ðn�1Þþ1; a; um�ðn�1Þþ2Þ � � � � � ~�ðum�n�1; a; um�nÞ;

1526 M. Demirci / Information Sciences 176 (2006) 1488–1530

we have

½WRðam�nÞ�ðu1; . . . ; um�nÞ 6 ½WRðam�ðn�1ÞÞ�ðu1; . . . ; um�ðn�1ÞÞ;

and this inequality obviously implies the next one

ð½WRðam�nÞ�ðu1; . . . ; um�nÞÞn�26 ð½WRðam�ðn�1ÞÞ�ðu1; . . . ; um�ðn�1ÞÞÞn�2

: ðB:3Þ

Now using the inequalities (B.1), (B.2) and (B.3), we observe that

ð½WRðam�nÞ�ðu1; . . . ; um�nÞÞn�1 � ½WRððumÞnÞ�ðv1; . . . ; vnÞ � ð½WRðamÞ�

ðu1; . . . ; umÞÞn�1

¼ ½ð½WRðam�nÞ�ðu1; . . . ; um�nÞÞn�2 � ½WRðam�nÞ�ðu1; . . . ; um�nÞ� � ½WRððumÞnÞ�

ðv1; . . . ; vnÞ � ð½WRðamÞ�ðu1; . . . ; umÞÞn�1

6 ½ð½WRðam�ðn�1ÞÞ�ðu1; . . . ; um�ðn�1ÞÞÞn�2 � ½WRðam�nÞ�ðu1; . . . ; um�nÞ�

� ½WRððumÞnÞ�ðv1; . . . ; vnÞ � ð½WRðamÞ�ðu1; . . . ; umÞÞn�1

¼ ½ð½WRðam�ðn�1ÞÞ�ðu1; . . . ; um�ðn�1ÞÞÞn�2 � ½WRðam�nÞ�ðu1; . . . ; um�nÞ�

� ð½WRððumÞn�1Þ�ðv1; . . . ; vn�1Þ � ~�ðvn�1; um; vnÞÞ � ½ð½WRðamÞ�

ðu1; . . . ; umÞÞn�2 � ½WRðamÞ�ðu1; . . . ; umÞ�

¼ ½ð½WRðam�ðn�1ÞÞ�ðu1; . . . ; um�ðn�1ÞÞÞn�2 � ½WRððumÞn�1Þ�ðv1; . . . ; vn�1Þ

� ð½WRðamÞ�ðu1; . . . ; umÞÞn�2� � ð½WRðam�nÞ�ðu1; . . . ; um�nÞ� ½WRðamÞ�ðu1; . . . ; umÞÞ � ~�ðvn�1; um; vnÞ6 Eðum�ðn�1Þ; vn�1Þ � Eðum�ðn�1ÞDum; um�nÞ � ~�ðvn�1; um; vnÞ¼ ½Eðum�ðn�1Þ; vn�1Þ � ~�ðvn�1; um; vnÞ� � Eðum�ðn�1ÞDum; um�nÞ6 ~�ðum�ðn�1Þ; um; vnÞ � Eðum�ðn�1ÞDum; um�nÞ6 Eðum�ðn�1ÞDum; vnÞ � Eðum�ðn�1ÞDum; um�nÞ 6 Eðum�n; vnÞ

and hence the assertion follows. In case of the condition (b), proof of the

claimed inequality is analogous to [3, Lemma 4.1(i)], and is skipped here. h

Appendix C. Proof of Lemma 3.7

Since the desired inequality is clear for u1 5 a or u�1 5 b, it is sufficient to

study on the assumption u1 = a and u�1 = b. In a similar way to Lemma 3.5, we

only state the proof for only the case (a). For n = 1, we have

½WRðaÞ�ðu1Þ � ½WRða�1Þ�ðe;bÞðu�1Þ ¼ 1:

M. Demirci / Information Sciences 176 (2006) 1488–1530 1527

Since b 2 InvðX ;~�Þða; eÞ and c 2 InvðX ;~�Þða; e0Þ, we get from Proposition 2.2(iii)

that b �E c, i.e. E(b,c) = E(c,u�1) = 1, so the claim is obvious for n = 1. Let

us now deal with the case n P 2. The condition (VAS) of vague associativity

allows us to put down the following set of inequalities:

~�ðun; b; un�1Þ ¼ ~�ðun; b; un�1Þ � ~�ðc; un�1; cDun�1Þ � ~�ðc; un; e0Þ � ~�ðe0; b; bÞ6 EðcDun�1; bÞ ¼ ~�ðc; un�1; bÞ;

~�ðun�1; b; un�2Þ � ~�ðc; un�1; bÞ � ~�ðb; b; u�2Þ¼ ~�ðun�1; b; un�2Þ � ~�ðc; un�2; cDun�2Þ � ~�ðc; un�1; bÞ� ~�ðb; b; u�2Þ6 EðcDun�2; un�2Þ ¼ ~�ðc; un�2; u�2Þ;

~�ðun�2; b; un�3Þ � ~�ðc; un�2; u�2Þ � ~�ðu�2; b; u�3Þ¼ ~�ðun�2; b; un�3Þ � ~�ðc; un�3; cDun�3Þ � ~�ðc; un�2; u�2Þ � ~�ðu�2; b; u�3Þ6 EðcDun�3; u�3Þ ¼ ~�ðc; un�3; u�3Þ;

..

.

~�ðu2; b; u1Þ � ~�ðc; u2; u�ðn�2ÞÞ � ~�ðu�ðn�2Þ; b; u�ðn�1ÞÞ¼ ~�ðu2; b; u1Þ � ~�ðc; u1; cDu1Þ � ~�ðc; u2; u�ðn�2ÞÞ � ~�ðu�ðn�2Þ; b; u�ðn�1ÞÞ6 EðcDu1; u�ðn�1ÞÞ ¼ ~�ðc; u1; u�ðn�1ÞÞ:

Now exploiting this set of inequalities, we observe that

~�ðc; u1; u�ðn�1ÞÞP ~�ðu2; b; u1Þ � ~�ðu�ðn�2Þ; b; u�ðn�1ÞÞ � ~�ðc; u2; u�ðn�2ÞÞP ~�ðu2; b; u1Þ � ~�ðu�ðn�2Þ; b; u�ðn�1ÞÞ � ½~�ðu3; b; u2Þ� ~�ðu�ðn�3Þ; b; u�ðn�2ÞÞ � ~�ðc; u3; u�ðn�3ÞÞ�¼ ½~�ðu3; b; u2Þ � ~�ðu2; b; u1Þ� � ½~�ðu�ðn�3Þ; b; u�ðn�2ÞÞ� ~�ðu�ðn�2Þ; b; u�ðn�1ÞÞ� � ~�ðc; u3; u�ðn�3ÞÞ

P ½~�ðu3; b; u2Þ � ~�ðu2; b; u1Þ� � ½~�ðu�ðn�3Þ; b; u�ðn�2ÞÞ � ~�ðu�ðn�2Þ; b; u�ðn�1ÞÞ�� ½~�ðu4; b; u3Þ � ~�ðu�ðn�4Þ; b; u�ðn�3ÞÞ � ~�ðc; u4; u�ðn�4ÞÞ�¼ ½~�ðu4; b; u3Þ � ~�ðu3; b; u2Þ � ~�ðu2; b; u1Þ� � ½~�ðu�ðn�4Þ; b; u�ðn�3ÞÞ� ~�ðu�ðn�3Þ; b; u�ðn�2ÞÞ � ~�ðu�ðn�2Þ; b; u�ðn�1ÞÞ� � ~�ðc; u4; u�ðn�4ÞÞ

..

.

P ½~�ðun�1; b; un�2Þ � � � � � ~�ðu2; b; u1Þ� � ½~�ðu�1; b; u�2Þ � � � �� ~�ðu�ðn�2Þ; b; u�ðn�1ÞÞ� � ~�ðc; un�1; u�1Þ:

1528 M. Demirci / Information Sciences 176 (2006) 1488–1530

Thus we have

~�ðc; u1; u�ðn�1ÞÞP ½~�ðun�1; b; un�2Þ � � � � � ~�ðu2; b; u1Þ�� ½~�ðu�1; b; u�2Þ � � � � � ~�ðu�ðn�2Þ; b; u�ðn�1ÞÞ�� ~�ðc; un�1; u�1Þ: ðC:1Þ

Furthermore, the equality (2.1) gives that

~�ðc; un�1; u�1Þ ¼ ~�ðun; u�1; un�1Þ ¼ ~�ðun; b; un�1Þ:Therefore, owing to the inequality (C.1), we can write

~�ðc; u1; u�ðn�1ÞÞP ½~�ðun�1; b; un�2Þ � � � � � ~�ðu2; b; u1Þ�� ½~�ðu�1; b; u�2Þ � � � � � ~�ðu�ðn�2Þ; b; u�ðn�1ÞÞ�

�~�ðc; un�1; u�1Þ ¼ ½~�ðun; b; un�1Þ � � � � � ~�ðu2; b; u1Þ�� ½~�ðu�1; b; u�2Þ � � � � � ~�ðu�ðn�2Þ; b; u�ðn�1ÞÞ�

¼ ½WRðanÞ�ðu1; . . . ; unÞ � ½WRða�ðn�1ÞÞ�ðe;bÞðu�1; . . . ; u�ðn�1ÞÞ:ðC:2Þ

On the other hand, in virtue of (2.2), we have

~�ðc; u1; u�ðn�1ÞÞ ¼ ~�ðc; a; u�ðn�1ÞÞ ¼ ~�ðu�ðn�1Þ; b; cÞ: ðC:3ÞHence, multiplying both sides of the inequality (C.2) by ~�ðu�ðn�1Þ; b; u�nÞ

w.r.t. *, and using the equality (C.3), we observe that

½WRðanÞ�ðu1; . . . ; unÞ � ½WRða�nÞ�ðe;bÞðu�1; . . . ; u�nÞ

¼ ½WRðanÞ�ðu1; . . . ; unÞ � ð½WRða�ðn�1ÞÞ�ðe;bÞðu�1; . . . ; u�ðn�1ÞÞ� ~�ðu�ðn�1Þ; b; u�nÞÞ

6 ~�ðc; u1; u�ðn�1ÞÞ � ~�ðu�ðn�1Þ; b; u�nÞ ¼ ~�ðu�ðn�1Þ; b; cÞ � ~�ðu�ðn�1Þ; b; u�nÞ6 Eðc; u�nÞ:

Appendix D. Proof of Lemma 3.8

It is sufficient to prove the properties (i) and (ii) for only the case (a). The

proof for the case (b) is similar to [3, Lemma 4.1(i)], so it is omitted here.

(i) If u�1 5 b or u1 5 a or v�1 5 c, the considered inequality is trivial. Let

us assume that u�1 = b and u1 = a and v�1 = c. It is easy to infer from Lemma

3.7 that

ð½WRðamÞ�ðu1; . . . ; umÞÞnþ1 � ð½WRða�mÞ�ðe;bÞðu�1; . . . ; u�mÞÞnþ1

6 ½Eðc; u�mÞ�nþ1: ðD:1Þ

M. Demirci / Information Sciences 176 (2006) 1488–1530 1529

Furthermore, if we consider Proposition 2.2(vi) and use Lemma 2.8(i), then we

possess the inequality

½WRððumÞ�nÞ�ðe0;cÞðc; v�2 . . . ; v�nÞ � ½Eðc; u�mÞ�nþ1

6 ½WRððu�mÞnÞ�ðu�m; v�2 . . . ; v�nÞ: ðD:2Þ

On the other hand, considering Remark 3.6, we directly obtain from Lemma

3.5 the inequality

ð½WRða�m�nÞ�ðe;bÞðu�1; . . . ; u�m�nÞÞn�1 � ½WRððu�mÞnÞ�ðv�1; . . . ; v�nÞ

� ð½WRða�mÞ�ðe;bÞðu�1; . . . ; u�mÞÞn�1

6 Eðu�m�n; v�nÞ: ðD:3Þ

Now exploiting the inequalities (D.1), (D.2) and (D.3), we observe that

ð½WRða�m�nÞ�ðe;bÞðu�1; . . . ; u�m�nÞÞn�1 � ð½WRðamÞ�ðu1; . . . ; umÞÞnþ1

� ð½WRða�mÞ�ðe;bÞðu�1; . . . ; u�mÞÞ2n � ½WRððumÞ�nÞ�ðe0;cÞðv�1; . . . ; v�nÞ

¼ ð½WRða�m�nÞ�ðe;bÞðu�1; . . . ; u�m�nÞÞn�1 � ð½WRðamÞ�ðu1; . . . ; umÞÞnþ1

� ½ð½WRða�mÞ�ðe;bÞðu�1; . . . ; u�mÞÞnþ1 � ð½WRða�mÞ�ðe;bÞðu�1; . . . ; u�mÞÞn�1�

� ½WRððumÞ�nÞ�ðe0 ;cÞðv�1; . . . ; v�nÞ

¼ ð½WRða�m�nÞ�ðe;bÞðu�1; . . . ; u�m�nÞÞn�1 � ½ð½WRðamÞ�ðu1; . . . ; umÞÞnþ1

� ð½WRða�mÞ�ðe;bÞðu�1; . . . ; u�mÞÞnþ1� � ð½WRða�mÞ�ðe;bÞðu�1; . . . ; u�mÞÞn�1

� ½WRððumÞ�nÞ�ðe0 ;cÞðc; . . . ; v�nÞ

6 ð½WRða�m�nÞ�ðe;bÞðu�1; . . . ; u�m�nÞÞn�1 � ½Eðc; u�mÞ�nþ1

� ð½WRða�mÞ�ðe;bÞðu�1; . . . ; u�mÞÞn�1 � ½WRððumÞ�nÞ�ðe0 ;cÞðc; . . . ; v�nÞ

¼ ð½WRða�m�nÞ�ðe;bÞðu�1; . . . ; u�m�nÞÞn�1 � ð½WRððumÞ�nÞ�ðe0 ;cÞðc; . . . ; v�nÞ

� ½Eðc; u�mÞ�nþ1Þ � ð½WRða�mÞ�ðe;bÞðu�1; . . . ; u�mÞÞn�1

6 ð½WRða�m�nÞ�ðe;bÞðu�1; . . . ; u�m�nÞÞn�1 � ½WRððu�mÞnÞ�ðu�m; v�2 . . . ; v�nÞ

� ð½WRða�mÞ�ðe;bÞðu�1; . . . ; u�mÞÞn�1

6 Eðu�m�n; v�nÞ:

(ii) For the case u1 5 a or u�1 5 b or v�1 5 d, the assertion is obvious. As-

sume that u1 = a,u�1 = b and v�1 = d. Let b�1 be an inverse of b w.r.t. e, i.e.

b�1 2 InvðX ;~�Þðb; eÞ. Then by using (i) we can write

1530 M. Demirci / Information Sciences 176 (2006) 1488–1530

ð½WRðb�m�nÞ�ðe;b�1Þðb�1; u2; . . . ; um�nÞÞn�1 � ð½WRðbmÞ�ðu�1; . . . ; u�mÞÞnþ1

� ð½WRðb�mÞ�ðe;b�1Þðb�1; u2; . . . ; umÞÞ2n

� ½WRððu�mÞ�nÞ�ðe0 ;dÞðv�1; . . . ; v�nÞ 6 Eðum�n; v�nÞ: ðD:4Þ

On the other hand, we have from Proposition 2.2(ii) that E(a,b�1) = 1.

Then, applying Lemma 2.8(i), we obtain that

½WRðb�m�nÞ�ðe;b�1Þðb�1; u2; . . . ; um�nÞ¼ ½WRðb�m�nÞ�ðe;b�1Þðb

�1; u2; . . . ; um�nÞ � Eða; b�1Þ6 ½WRðam�nÞ�ða; u2 . . . ; um�nÞ¼ ½WRðam�nÞ�ða; u2 . . . ; um�nÞ � Eða; b�1Þ6 ½WRðb�m�nÞ�ðe;b�1Þðb

�1; u2; . . . ; um�nÞ;

i.e.

½WRðb�m�nÞ�ðe;b�1Þðb�1; u2; . . . ; um�nÞ ¼ ½WRðam�nÞ�ða; u2; . . . ; um�nÞ:

Similarly

½WRðb�mÞ�ðe;b�1Þðb�1; u2; . . . ; umÞ ¼ ½WRðamÞ�ða; u2; . . . ; umÞ:

Thus, using these two equalities in (D.4), the assertion becomes

straightforward.

References

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

equivalence relations, Part II: Vague algebraic notions, Int. J. General Syst. 32 (2) (2003) 157–

175.

[2] 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, Int. J. General Syst. 32 (2) (2003) 177–201.

[3] M. Demirci, The generalized associative law in vague groups and its applications—I, Inform Sci

(in press).

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

arithmetic, Fuzzy Sets Syst (accepted).

[5] T.W. Hungerford, Algebra, Holt, Rinehart and Winston Inc., New York, 1974.

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

Academic Press, San Diego, 1971.