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Information Sciences 174 (2005) 123–142

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Vague groups and generalizedvague subgroups on the basis of ([0,1], 6 ,^)

Sevda Sezer *

Department of Mathematics, Faculty of Science and Arts, Akdeniz University, 07058 Antalya, Turkey

Received 2 July 2004; received in revised form 21 July 2004; accepted 24 July 2004

Abstract

In this paper, various elementary properties of vague groups and some properties of

vague binary operations related with their associativity aspects are obtained. Further-

more, the concept of vague isomorphism is defined and some basic properties of this

concept are studied. The concept of external direct product of vague groups is estab-

lished. Later the definition of generalized vague subgroup, which is a generalization

of the vague subgroup defined by Demirci, is introduced, and the validity of some clas-

sical results in this setting is investigated on the basis of the particular integral commu-

tative, complete quasi-monoidal lattice ([0,1], 6 ,^).� 2004 Elsevier Inc. All rights reserved.

Keywords: Fuzzy equality; Fuzzy function; Vague group; External direct product of vague groups;

Vague homomorphism; Vague isomorphism; Vague subgroup; Generalized vague subgroup

0020-0255/$ - see front matter � 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.ins.2004.07.016

* Tel.: +90 242 3102360; fax: +90 242 2278911/3102360.

E-mail address: [email protected]

mailto:[email protected]

124 S. Sezer / Information Sciences 174 (2005) 123–142

1. Introduction

After the introduction of fuzzy sets by Zadeh [14], fuzzy settings of various

algebraic concepts were studied by several authors, using the approach of

Rosenfeld [11]. In Rosenfeld�s paper, only the subsets are fuzzy and the group

operation remains crisp. To get a more general extension, Demirci [2] definedthe concept of ‘‘vague group’’ based on fuzzy equalities and fuzzy functions

given in [1,3]. In the same paper, the concepts of vague subgroup and vague

homomorphism were defined and the validity of some classical results in this

setting were investigated. Later, some characterizations and examples concern-

ing vague groups were given by Demirci and Coker [4].

The general theory of vague algebraic notions as well as vague groups and

vague subgroups has been established in [6–8] based on a fixed integral commu-

tative, complete quasi-monoidal lattice (for short, cqm-lattice) (L, 6 ,w). Thiswork introduces some elementary properties of vague groups and generalized

vague subgroups, and establishes some new results on the basis of the parti-

cular integral, commutative cqm-lattice ([0,1], 6 ,^).

2. Preliminaries

The symbol ‘‘^’’ will always stand for the minimum operation betweenfinitely many real numbers, and X, Y, G will always stand for crisp and non-

empty sets in this paper.

Definition 1 [1]. A mapping EX : X · X ! [0,1] is called a fuzzy equality on X

if the following conditions are satisfied:

(E.1) EX ðx; yÞ ¼ 1 () x ¼ y, "(x,y) 2 X,

(E.2) EX(x,y) = EX(y,x), "x,y 2 X,(E.3) EX(x,y) ^ EX(y,z) 6 EX(x,z), "x,y,z 2 X.

For x,y 2 X, the real number EX(x,y) shows the degree of the equality of x

and y. One can always define a fuzzy equality on X with respect to (abbreviated

to ‘‘w.r.t.’’) the classical equality of the elements of X. Indeed, the mapping

EcX : X � X ! ½0; 1�, defined by

EcX ðx; yÞ :¼

1; x ¼ y;

0; x 6¼ y

�is obviously a fuzzy equality on X.

Definition 2 [5]. Let EX and EY be two fuzzy equalities on X and Y,

respectively. Then a fuzzy relation ~� from X to Y (i.e., a fuzzy subset ~� of

S. Sezer / Information Sciences 174 (2005) 123–142 125

X · Y) is called a strong fuzzy function from X to Y w.r.t. the fuzzy equalities

EX and EY, denoted by ~� : X,Y , if the characteristic function

l~� : X � Y ! ½0; 1� of ~� satisfies the following two conditions:

(F.1) For each x 2 X, there exists y 2 Y such that l~�ðx; yÞ ¼ 1,

(F.2) For each x1,x2 2 X, y1,y2 2 Y,

l~�ðx1; y1Þ ^ l~�ðx2; y2Þ ^ EX ðx1; x2Þ6EY ðy1; y2Þ:

The concepts of vague binary operation on X and transitivity of a vague bin-

ary operation are defined as follows.

Definition 3 [2,5]

(i) A strong fuzzy function ~� : X � X,X w.r.t. a fuzzy equality EX·X on

X · X and a fuzzy equality EX on X is called a vague binary operation

on X w.r.t. EX ·X and EX. (For all (x1,x2) 2 X · X, x3 2 X,

l~�ððx1; x2Þ; x3Þ will be denoted by l~�ðx1; x2; x3Þ for the sake of simplicity.)

(ii) A vague binary operation ~� on X w.r.t. EX·X and EX is said to be tran-

sitive of the first order if l~�ða; b; cÞ ^ EX ðc; dÞ6 l~�ða; b; dÞ for all

a,b,c,d 2 X.(iii) A vague binary operation ~� on X w.r.t. EX·X and EX is said to be tran-

sitive of the second order if l~�ða; b; cÞ ^ EX ðb; dÞ6 l~�ða; d; cÞ for all

a,b,c,d 2 X.

(iv) A vague binary operation ~� on X w.r.t. EX·X and EX is said to be tran-

sitive of the third order if l~�ða; b; cÞ ^ EX ða; dÞ6 l~�ðd; b; cÞ for all

a,b,c,d 2 X.

3. Vague groups

3.1. Definition of vague groups

The concept of vague group, which is the base of this work, is given by the

following definition.

Definition 4 [2]. Let ~� be a vague binary operation on G w.r.t. a fuzzy equality

EG·G on G · G and a fuzzy equality EG on G. Then

(i) G together with ~�, denoted by hG; ~�;EG�G;EGi or simply hG; ~�i, is called a

vague semigroup if the characteristic function l~� : G� G� G ! ½0; 1� of~� fulfills the condition: For all a,b,c,d,m,q,w 2 G,


l~�ðb; c; dÞ ^ l~�ða; d;mÞ ^ l~�ða; b; qÞ ^ l~�ðq; c;wÞ6EGðm;wÞ:(ii) A vague semigroup hG; ~�i is called a vague monoid if there exists a two-

sided identity element e 2 G, that is an element e satisfyingl~�ðe; a; aÞ ^ l~�ða; e; aÞ ¼ 1 for each a 2 G.

(iii) A vague monoid hG; ~�i is called a vague group if for each a 2 G, there

exists a two-sided inverse element a�1 2 G, that is an element a�1 satis-

fying l~�ða�1; a; eÞ ^ l~�ða; a�1; eÞ ¼ 1.

(iv) A vague semigroup hG; ~�i is said to be commutative (Abelian) if

l~�ða; b;mÞ ^ l~�ðb; a;wÞ6EGðm;wÞ for each a,b,m,w 2 G.

In particular, if ~� is a vague binary operation on G w.r.t. EcG�G on G · G and

EcG on G such that l~�ðG� G� GÞ � f0; 1g, then a vague group hG; ~�i

corresponds in a one-to-one way with a group in the classical sense. In this

case, a vague group is simply called a crisp group. For a given classical group

hG,�i, it is known that an infinite number of nontrivial vague groups can be

defined on G [2,4,5].

In the rest of this paper, the notation hG; ~�i always stands for the vague

group hG; ~�i w.r.t. a fuzzy equality EG·G on G · G and a fuzzy equality EG

on G.

Proposition 5 [2]. For a given vague group hG; ~�i, there exists a unique binary

operation in the classical sense, denoted by �, on G such that hG,�i is a group in

the classical sense.

The binary operation ‘‘�’’ in Proposition 5 is explicitly given by the

equivalence

a � b :¼ c () l~�ða; b; cÞ ¼ 1; 8a; b; c 2 G: ð1ÞThe binary operation ‘‘�’’, defined by the equivalence (1), is called the ordinary

description of ~�, and is denoted by � ¼ ordð~�Þ in [5,7,8].

If ~� is a vague binary operation on G w.r.t. a fuzzy equality EG·G on G · G

and a fuzzy equality EG on G, in the rest of this paper the ordinary description

of ~� will be denoted by �. In this case, from [5,7] we have the followingproperty:

l~�ða; b; a � bÞ ¼ 1 and l~�ða; b; cÞ6EGða � b; cÞ; 8a; b; c 2 G: ð2ÞIf hG; ~�i is a vague semigroup, then hG,�i is a semigroup [5].

Conditions under which the converse holds are given below.

Proposition 6. Let ~� be a vague binary operation on G w.r.t. EG·G and EG. IfhG,�i is a semigroup and ~� is transitive of the second and third orders, then hG; ~�iis a vague semigroup.


Proof. If hG,�i is a semigroup, then a � (b � c) = (a � b) � c for each a,b,c 2 G.

Since ~� is transitive of the second and third orders then for all

a,b,c,d,m,q,w 2 G we have

l~�ðb; c; dÞ ^ l~�ða; d;mÞ ^ l~�ða; b; qÞ ^ l~�ðq; c;wÞ6EGðb � c; dÞ ^ l~�ða; d;mÞ ^ EGða � b; qÞ ^ l~�ðq; c;wÞ6l~�ða; b � c;mÞ ^ l~�ða � b; c;wÞ6EGða � ðb � cÞ;mÞ ^ EGðða � bÞ � c;wÞ6EGðm;wÞ:

Hence, hG; ~�i is a vague semigroup. h

If ~� is a vague binary operation on G, hG,�i is a semigroup and ~� is not tran-sitive of the second and third orders then hG; ~�i may not be a vague semigroup.

The following example illustrates this case.

Example 7. Take the set G :¼ {0,1,2}, and consider the fuzzy equality

EG = [EG(i, j)]3·3 on G by means of the matrix

EG(i, j) 0 1 2

0 1 .5 .41 .5 1 .4

2 .4 .4 1

and the fuzzy equality EG�G :¼ EcG�G on G · G. Now let us define the fuzzy

function ~� : G� G,G by the matrix l~� ¼ ½l~�ðx; i; jÞ�3�3�3, where x, i, j = 0,1,2

(i: rows, j: columns) and

0 1 2

l~�ð0; i; jÞ0 1 .5 .4

1 .5 1 .4

2 .4 .4 1

l~�ð1; i; jÞ0 .3 1 .21 .1 .1 1

2 1 .1 .2

l~�ð2; i; jÞ0 .3 .3 1

1 1 .2 .1

2 .2 1 .2


Under these selections, ~� is a vague binary operation on G and we get hG,�i as

� 0 1 2

0 0 1 21 1 2 1

2 2 0 1

Since

l~�ð1; 0; 1Þ ^ EGð0; 1Þ ¼ 1 ^ :5 ¼ :5il~�ð1; 1; 1Þ ¼ :1

and

l~�ð0; 0; 0Þ ^ EGð0; 1Þ ¼ 1 ^ :5 ¼ :5il~�ð1; 0; 0Þ ¼ :3;

~� is not transitive of the second and third orders on G. Furthermore, hG; ~�i alsois not a vague semigroup because

l~�ð0; 1; 0Þ ^ l~�ð0; 0; 0Þ ^ l~�ð0; 0; 1Þ ^ l~�ð1; 1; 2Þ ¼ :5iEGð0; 2Þ ¼ :4:

3.2. Some results on transitivity

The following theorem states that some results of classical algebra are also

valid for the vague algebra.

Theorem 8. Let hG; ~�i be a vague group and e an identity element of hG; ~�i.

(a) If the vague binary operation ~� is transitive of the third order, then

(i) EG(x � a,b) = EG(x,b � a�1) and EG(x � a, e) = EG(x,a�1) for all

a,b,x 2 G.

(ii) EG(a,b) = EG(a � x,b � x) for all a,b,x 2 G.

(b) If the vague binary operation ~� is transitive of the second order, then

(i) EG(a � y,b) = EG(y,a�1 � b) and EG(a � y,e) = EG(y,a

�1) for all

a,b,y 2 G.

(ii) EG(a,b) = EG(y � a,y � b) for all a,b,y 2 G.

(c) If the vague binary operation ~� is transitive of the second and third orders,

then EG(a,b)^EG(c,d) 6 EG(a � c,b � d) for all a,b, c,d 2 G.

Proof

(a) (i) By the third-order transitivity of ~�, for all a,b,x 2 G we have

EGðx � a; bÞ ¼ EGðx � a; bÞ ^ l~�ðx � a; a�1; xÞ6l~�ðb; a�1; xÞ6EGðb � a�1; xÞ:


On the other hand,

EGðb � a�1; xÞ ¼ EGðb � a�1; xÞ ^ l~�ðb � a�1; a; bÞ6 l~�ðx; a; bÞ6EGðx � a; bÞ;

i.e., EG(x � a,b) = EG(x,b � a�1). In particular, if b = e then, we observe

that EG(x � a,e) = EG(x,a�1).

(ii) By the third-order transitivity of ~�, we have

EGða; bÞ ¼ EGða; bÞ ^ l~�ðb; x; b � xÞ6 l~�ða; x; b � xÞ6EGða � x; b � xÞand

EGða � x; b � xÞ ¼ EGða � x; b � xÞ ^ l~�ðb � x; x�1; bÞ6l~�ða � x; x�1; bÞ6EGða; bÞ;

for all a,b,x 2 G. So EG(a,b) = EG(a � x,b � x).

(b) (i) By the second-order transitivity of ~�, for all a,b,y 2 G we have

EGða � y; bÞ ¼ EGða � y; bÞ ^ l~�ða�1; a � y; yÞ6 l~�ða�1; b; yÞ6EGðy; a�1 � bÞ:

On the other hand,

EGðy; a�1 � bÞ ¼ EGðy; a�1 � bÞ ^ l~�ða; a�1 � b; bÞ6 l~�ða; y; bÞ6EGða � y; bÞ;

i.e., EG(y,a�1 � b) = EG(a � y,b).

In particular, if b = e, then we have EG(a � y,e) = EG(y,a�1).

(ii) By the second-order transitivity of ~�, we have the inequalities

EGða; bÞ ¼ EGða; bÞ ^ l~�ðy; b; y � bÞ6 l~�ðy; a; y � bÞ6EGðy � a; y � bÞ;

EGðy � a; y � bÞ ¼ EGðy � a; y � bÞ ^ l~�ðy�1; y � b; bÞ6 l~�ðy�1; y � a; bÞ6EGða; bÞ

for all a,b,y 2 G, thus EG(a,b) = EG(y � a,y � b).(c) By the second and third order transitivity of ~�, we have

EGða; bÞ ^ EGðc; dÞ ¼ l~�ða; c; a � cÞ ^ EGða; bÞ ^ l~�ðb; d; b � dÞ ^ EGðc; dÞ6 l~�ðb; c; a � cÞ ^ l~�ðb; c; b � dÞ6EGða � c; b � dÞ; 8a; b; c; d 2 G: �

Some inequalities about associative properties of vague semigroups are ob-

tained in the following two propositions.

Proposition 9. Let ~� be a vague binary operation on G w.r.t. fuzzy equalities

EG·G on G · G and EG on G such that ~� is transitive of the third order. For

� ¼ ordð~�Þ, let hG,�i be a semigroup. For n P 2, a1,a2, . . ., an, u1,u2, . . ., un 2 G

and u1 = a1, the following inequality is satisfied:


n̂�1

i¼1

l~�ðui; aiþ1; uiþ1Þ6EGða1 � a2 � � � � � an; unÞ:

Proof. To prove the assertion, we apply induction on n. From the inequality

(2), we observe that l~�ðu1; a2; u2Þ6EGðu1 � a2; u2Þ ¼ EGða1 � a2; u2Þ, so the

required inequality is true for n = 2. We assume that the required inequality

is true for n � 1, i.e.

n̂�2

i¼1

l~�ðui; aiþ1; uiþ1Þ6EGða1 � a2 � � � � � an�1; un�1Þ:

Using this inequality and the third order transitivity of ~�, we obtain the

inequalities

n̂�1

i¼1

l~�ðui; aiþ1; uiþ1Þ

¼n̂�2

i¼1

l~�ðui; aiþ1; uiþ1Þ !

^ l~�ðun�1; an; unÞ

6EGða1 � a2 � � � � � an�1; un�1Þ ^ l~�ðun�1; an; unÞ6l~�ða1 � a2 � � � � � an�1; an; unÞ6EGða1 � a2 � � � � � an; unÞ: �

Proposition 10. Let ~� be a vague binary operation on G w.r.t. fuzzy equalities

EG·G on G · G and EG on G such that ~� is transitive of the second order. Let

� ¼ ordð~�Þ and hG,� i be a semigroup. For n P 2, a1,a2, . . ., an, u1,u2, . . ., un 2 G

and u1 = a1, the following inequality is satisfied:

n̂�1

i¼1

l~�ðaiþ1; ui; uiþ1Þ6EGðan � an�1 � � � � � a1; unÞ: ð3Þ

Proof. In a similar fashion to Proposition 9, the assertion can be easily verified

by applying induction on n, and the details are omitted. h

3.3. External direct product of vague groups

The aim of this subsection is to establish the concept of external direct prod-

uct of vague groups. First, let us introduce the concept of the product fuzzy

equalities.

Let EGj and EGj � EGj be fuzzy equalities on Gj and Gj · Gj, respectively, for

j = 1,2, . . .,n. In this case it is easily seen that the maps, defined by


EG1��Gnððx1; . . . ; xnÞ; ðy1; . . . ; ynÞÞ :¼n̂

j¼1

EGjðxj; yjÞ ð4Þ

and

EðG1��GnÞ�ðG1��GnÞðððx1; . . . ; xnÞ; ðy1; . . . ; ynÞÞ; ððz1; . . . ; znÞ; ðt1; . . . ; tnÞÞÞ

:¼n̂

j¼1

EGj�Gjððxj; yjÞ; ðzj; tjÞÞ

ð5Þare fuzzy equalities on G1 · � � � · Gn and (G1 · � � � · Gn) · (G1 · � � � · Gn),respectively.

Proposition 11. For each j = 1,2, . . ., n, let hGj; ~�ji be a vague group w.r.t. the

fuzzy equalities EGj�Gj on Gj · Gj and EGj on Gj. Let the fuzzy relation ~� :ðG1 � � � � � GnÞ � ðG1 � � � � � GnÞ,G1 � � � � � Gn be defined by

l~�ðða1; . . . ; anÞ; ðb1; . . . ; bnÞ; ðc1; . . . ; cnÞÞ :¼n̂

j¼1

l~�jðaj; bj; cjÞ:

For the fuzzy equalities EðG1��GnÞ and EðG1��GnÞ�ðG1��GnÞ defined by (4) and

(5), respectively, hG1 � � � � � Gn; ~�i is a vague group w.r.t. EðG1��GnÞ�ðG1��GnÞand EðG1��GnÞ.

Proof. Let xj,yj 2 Gj and �j ¼ ordð~�jÞ for all j = 1,2, . . .,n. Therefore,

l~�ððx1; . . . ; xnÞ; ðy1; . . . ; ynÞ; ðx1�1y1; . . . ; xn�nynÞÞ ¼n̂

j¼1

l~�jðxj; yj; xj�jyjÞ ¼ 1;

i.e., the condition (F.1) is satisfied. We define

R :¼ l~�ðða1; . . . ; anÞ; ðb1; . . . ; bnÞ; ðc1; . . . ; cnÞÞ;

S :¼ l~�ððx1; . . . ; xnÞ; ðy1; . . . ; ynÞ; ðz1; . . . ; znÞÞ;

T :¼ EðG1��GnÞ�ðG1��GnÞððða1; . . . ; anÞ; ðb1; . . . ; bnÞÞ;ððx1; . . . ; xnÞ; ðy1; . . . ; ynÞÞÞ:

Then, for each j 2 J, we have

R ^ S ^ T 6l~�jðaj; bj; cjÞ ^ l~�jðxj; yj; zjÞ ^ EGj�Gjððaj; bjÞ; ðxj; yjÞÞ6EG ðcj; zjÞ:
j


Thus

R ^ S ^ T 6

n̂

j¼1

EGjðcj; zjÞ ¼ EG1��Gnððc1; . . . ; cnÞ; ðz1; . . . ; znÞÞ;

i.e., the condition (F.2) is satisfied. Thus, ~� is a vague binary operation on

G1 · � � � · Gn.

Let a = (a1, . . .,an), b = (b1, . . .,bn), c = (c1, . . .,cn), d = (d1, . . .,dn), q = (q1, . . .,qn), m = (m1, . . .,mn), w = (w1, . . .,wn) and

Y :¼ l~�ðb; c; dÞ ^ l~�ða; d;mÞ ^ l~�ða; b; qÞ ^ l~�ðq; c;wÞ:Then, for all j 2 J, we have

Y 6 l~�jðbj; cj; djÞ ^ l~�jðaj; dj;mjÞ ^ l~�jðaj; bj; qjÞ ^ l~�jðqj; cj;wjÞ6EGjðmj;wjÞ:

Therefore, we get

Y 6

n̂

j¼1

EGjðmj;wjÞ ¼ EG1��Gnððm1; . . . ;mnÞ; ðw1; . . . ;wnÞÞ:

If ej is the identity element of hGj; ~�ji, then

l~�ððx1; . . . ; xnÞ; ðe1; . . . ; enÞ; ðx1; . . . ; xnÞÞ ¼n̂

j¼1

l~�jðxj; ej; xjÞ ¼ 1

and

l~�ððe1; . . . ; enÞ; ðx1; . . . ; xnÞ; ðx1; . . . ; xnÞÞ

¼n̂

j¼1

l~�iðej; xj; xjÞ ¼ 1 for all j ¼ 1; . . . ; n:

Hence (e1, . . .,en) is the identity element of hG1 � � � � � Gn; ~�i. On the other

hand, we can write

l~�ððx1; . . . ; xnÞ; ðx�11 ; . . . ; x�1

n Þ; ðe1; . . . ; enÞÞ ¼n̂

j¼1

l~�jðxj; x�1j ; ejÞ ¼ 1

and

l~�ððx�11 ; . . . ; x�1

n Þ; ðx1; . . . ; xnÞ; ðe1; . . . ; enÞÞ ¼n̂

j¼1

l~�jðx�1j ; xj; ejÞ ¼ 1;

i.e. (x1�1, . . .,xn

�1) is the inverse of (x1, . . .,xn) in hG1 � � � � � Gn; ~�i. Thus,

hG1 � � � � � Gn; ~�i is a vague group w.r.t. EðG1��GnÞ�ðG1��GnÞ and

EG1��Gn . h


The vague group, obtained in Proposition 11, will be called the external

direct product of the vague groups hGj; ~�ji. It is obvious that the external directproduct group of classical groups hGj,�ji, denoted by hG1 · � � � · Gn,�i, coin-cides with hG1 · � � � · Gn,•i.

For nonempty sets X, Y and Z with Z = X · Y, let hZ; ~�i be a vague group

w.r.t. fuzzy equalities EZ·Z on Z · Z and EZ on Z. The identity element ofhZ; ~�i and the inverse element of (x,y) 2 Z are denoted by eZ = (eX,eY) and

(x,y)�1 = (x�1,y�1), respectively.

If we define

EX ðx1; x2Þ :¼ EZððx1; eY Þ; ðx2; eY ÞÞ;

EX�X ððx1; x2Þ; ðy1; y2ÞÞ :¼ EZ�Zðððx1; eY Þ; ðx2; eY ÞÞ; ððy1; eY Þ; ðy2; eY ÞÞÞ;

EY ðy1; y2Þ :¼ EZððeX ; y1Þ; ðeX ; y2ÞÞand

EY�Y ððx1; x2Þ; ðy1; y2ÞÞ :¼ EZ�ZðððeX ; x1Þ; ðeX ; x2ÞÞ; ððeX ; y1Þ; ðeX ; y2ÞÞÞ;then it can be easily checked that EX, EY, EX·X and EY·Y are fuzzy equalities on

X, Y, X · X and Y · Y, respectively. These fuzzy equalities will be called the

projections of EZ on X and Y and of EZ·Z on X · X and Y · Y, respectively.

Proposition 12. For nonempty set X, Y and Z with Z = X · Y, let hZ; ~�i be a

vague group w.r.t. fuzzy equalities EZ·Z on Z · Z and EZ on Z. If

~�1 : X � X,X such that l~�1ðx1; x2; tÞ :¼ l~�ððx1; eY Þ; ðx2; eY Þ; ðt; eY ÞÞand

~�2 : Y � Y,Y such that l~�2ðy1; y2;wÞ :¼ l~�ððeX ; y1Þ; ðeX ; y2Þ; ðeX ;wÞÞare vague binary operations on X and Y, then for the projections EX, EY of EZ on

X and Y and for the projections EX·X, EY·Y of EZ·Z on X · X and Y · Y, respec-

tively, hX ; ~�1i is a vague group w.r.t. the fuzzy equalities EX·X on X · X and EX

on X. Also hY ; ~�2i is a vague group w.r.t. the fuzzy equalities EY·Y on Y · Y and

EY on Y.

Proof. For all a,b,c,d,m,q,w 2 X we have

l~�1ðb; c; dÞ ^ l~�1ða; d;mÞ ^ l~�1ða; b; qÞ ^ l~�1ðq; c;wÞ¼ l~�ððb; eY Þ; ðc; eY Þ; ðd; eY ÞÞ ^ l~�ðða; eY Þ; ðd; eY Þ; ðm; eY ÞÞ^ l~�ðða; eY Þ; ðb; eY Þ; ðq; eY ÞÞ ^ l~�ððq; eY Þ; ðc; eY Þ; ðw; eY ÞÞ

6EZððm; eY Þ; ðw; eY ÞÞ ¼ EX ðm;wÞ;

i.e., hX ; ~�1i is a vague semigroup. Furthermore, since (eX,eY) is an identity ele-

ment of hZ; ~�i and ðeX ; eY Þ�1 ¼ ðe�1X ; e�1

Y Þ ¼ ðeX ; eY Þ, we have


l~�1ðx; eX ; xÞ ¼ l~�ððx; eY Þ; ðeX ; eY Þ; ðx; eY ÞÞ ¼ 1

¼ l~�ððeX ; eY Þ; ðx; eY Þ; ðx; eY ÞÞ ¼ l~�1ðeX ; x; xÞ

and

l~�1ðx; x�1; eX Þ ¼ l~�ððx; eY Þ; ðx�1; eY Þ; ðeX ; eY ÞÞ

¼ l~�ððx; eY Þ; ðx; eY Þ�1; ðeX ; eY ÞÞ ¼ 1 ¼ l~�1ðx�1; x; eX Þ:

Hence, hX ; ~�1i is a vague group w.r.t. the fuzzy equalities EX·X on X · X and

EX on X.The same statement for hY ; ~�2i can be obtained in a similar way. h

4. Generalized vague subgroups

In [4] the authors mention that the definition of vague subgroups given by

them is too much strong. This restricts the base of vague algebra. The introduc-

tion of generalized vague subgroup will provide a richer base for vague algebra,because it protects classical results on a wider algebraic structure. Thus, this

section provides an answer to some questions raised in [4]. 1

For a given fuzzy equality EG on G and for a crisp subset A of G, the restric-

tion of the mapping EG to A · A, denoted by EA, is obviously a fuzzy equality

on A.

Definition 13 [2]. Let ~� be a vague binary operation on G and A a crisp subset

of G. Then A is said to be vague closed under ~� if

l~�ða; b; cÞ ¼ 1 ) c 2 A; 8a; b 2 A; 8c 2 G:

For a given vague binary operation ~� on G w.r.t. EG·G and EG, if a crisp subset

A of G is vague closed under ~�, then it is not difficult to observe that ~�jA�A�A is a

vague binary operation on A, and ~�jA�A�A inherits the transitivity properties of~�.

Definition 14 [2]. Let hG; ~�i be a vague group, and let A be a nonempty and

crisp subset of G that is vague closed under ~�. Then A is said to be a vague

subgroup of G if hA; ~�jA�A�Ai is itself a vague group w.r.t. the fuzzy equalities

EA·A on A · A and EA on A.

1 A preliminary version of some results in this section was first presented at the 15th Turkish

National Mathematical Symposium Proceedings [13].


Definition 15 [2]. Let hG; ~�i and hH ; ~Hi be two vague semigroups. A function

(in the classical sense) U : G!H is called a vague homomorphism if

l~�ða; b; cÞ6 l ~HðUðaÞ;UðbÞ;UðcÞÞ; 8a; b; c 2 G.

Definition 16 [2]. Let hG; ~�i and hH ; ~Hi be two vague groups, and let

U : G! H be a vague homomorphism. The crisp set {g 2 G : U(g) = eH} iscalled the vague kernel of U, and is denoted by VKerU.

Definition 17 [2]. Let EG and EH be, respectively, fuzzy equalities on G and H.

A function g : G ! H is said to be vague injective w.r.t. EG and EH if

EH(g(a),g(b)) 6 EG(a,b),"a,b 2 G.

It can be easily seen that a vague injective function is automatically injective

in the classical sense.

Definition 18. Let hG; ~�i and hH ; ~Hi be two vague semigroups, and let

U : G! H be a vague homomorphism. U : G! H is called a vague isomor-

phism if U is injective and surjective, and U�1 : H ! G is a vague

homomorphism.

It can be easily seen that if U is a vague isomorphism from hG; ~�i to hH ; ~Hi,then U is a classical isomorphism from hG,�i to hH,wi. Conditions underwhich the converse holds are given below.

Proposition 19. Let hG; ~�i and hH ; ~Hi be two vague groups. If U : G! H is a

surjective, vague injective, vague homomorphism and ~� is transitive of the first

order, then U is a vague isomorphism.

Proof. Obvious from Proposition 4.10 in [2]. h

Corollary 20. Let hG; ~�i and hH ; ~Hi be two vague groups for which the vague

binary operations ~� and ~H are transitive of the first order. If U : G ! H is a

surjective homomorphism such that EG(a,b) = EH(U(a),U(b)) for all a,b 2 G,

then U : G ! H is a vague isomorphism.

Proof. Obvious from Theorem 5.13(ii) in [5] and Proposition 19. h

In the classical algebra we know that if U is an isomorphism then the inverseof U is also an isomorphism. It is obvious that this statement is true for the va-

gue algebra, i.e., if U is a vague isomorphism, then the inverse of U is a vague

isomorphism, too.

The following definition is a generalization of the notion of vague subgroup.

In the rest of this paper, we will use this new definition of vague subgroup.


Definition 21. Let hG; ~�i be a vague group and A be a nonempty, crisp subset

of G. Let ~ be a vague binary operation on A such that

l ~ða; b; cÞ6l~�ða; b; cÞ; 8a; b; c 2 A:

If hA; ~i is itself a vague group w.r.t. the fuzzy equalities EA·A on A · A and EA

on A, then hA; ~i is said to be a generalized vague subgroup of hG; ~�i, denotedby hA; ~i 6

v:s

hG; ~�i.If hA; ~�i is a vague subgroup of hG; ~�i with respect to Definition 14, since

hA; ~�i 6v:shG; ~�i, we observe that Definition 21 is indeed a generalization of

Definition 14.

In particular, if ~� is a vague binary operation on G w.r.t EcG�G on G · G and

EcG on G such that l~�ðG� G� GÞ � f0; 1g, then a generalized vague subgroup

hA; ~i of hG; ~�i with respect to Definition 21 corresponds in a one-to-one way

to a subgroup in the classical sense.

For a given vague group hG; ~�i, because of the uniqueness of the identity

and the inverse of an element of hG; ~�i, it can be easily seen that if

hA; ~i 6v:shG; ~�i, then the identity of hA; ~i and the inverse of x 2 A w.r.t. hA; ~i

are the identity of hG; ~�i and the inverse of x 2 A w.r.t. hG; ~�i, i.e., eA = eG and

x�1A ¼ x�1

G , respectively.

Example 22. Let G :¼ Z, A :¼ 2Z, a; b; c 2 R such that 0 6 c 6 b 6 a <1.We define

EZ : Z� Z ! ½0; 1�; EZðu; vÞ :¼1; u ¼ v;

a; u 6¼ v;

�

EZ�Z :¼ EcZ�Z,

E2Z : 2Z� 2Z ! ½0; 1�; E2Zðm; nÞ :¼ EZðm; nÞ;E2Z�2Z :¼ Ec

2Z�2Z,

~� : Z� Z,Z; l~�ðx; y; zÞ :¼1; xþ y ¼ z;

b; xþ y 6¼ z

�

and

~ : 2Z� 2Z,2Z; l ~ða; b; cÞ :¼1; aþ b ¼ c;

c; aþ b 6¼ c:

�

If x; y; z; u; v; t 2 Z, EZ�Zððx; yÞ; ðu; vÞÞ ¼ 1 and z 5 t, then either l~�ðx; y; zÞ 6¼ 1

or l~�ðx; y; tÞ 6¼ 1. Thus

l~�ðx; y; zÞ ^ l~�ðu; v; tÞ ^ EZ�Zððx; yÞ; ðu; vÞÞ6b6 a6EZðz; tÞ:


Since l~�ðx; y; xþ yÞ ¼ 1, ~� is a vague binary operation on Z. For each

a; b; c; d;m; q;w 2 Z such that m 5 w, since at least one of the equalities

b + c = d, a + d = m, a + b = q and q + c = w is not satisfied, we can write

l~�ðb; c; dÞ ^ l~�ða; d;mÞ ^ l~�ða; b; qÞ ^ l~�ðq; c;wÞ6 b6 a6EZðm;wÞ:

Thus, hZ; ~�i is a vague semigroup. For all x 2 Z, since l~�ðx; 0; xÞ ¼1 ¼ l~�ð0; x; xÞ and l~�ðx;�x; 0Þ ¼ 1 ¼ l~�ð�x; x; 0Þ, it follows that hZ; ~�i is a va-

gue group.

In a similar fashion to hZ; ~�i, it can be easily seen that h2Z; ~i is a vague

group. Using Definition 21, we obtain h2Z; ~i 6v:shZ; ~�i.

Proposition 23. Let hG; ~�i be a vague group. If hA; ~i 6v:shG; ~�i and

hB; ~�i 6v:shA; ~i, then hB; ~�i 6

v:shG; ~�i.

Proof. From the hypothesis, B � A � G and l~�ðx; y; zÞ6 l ~ðx; y; zÞ6 l~�ðx; y; zÞfor each x,y,z 2 B. Since hB; ~�i is a vague group, hB; ~�i 6

v:shG; ~�i. h

Proposition 24. Let hG; ~�i be a vague group, and let hA; ~i 6v:shG; ~�i,

hB; ~�i 6v:shG; ~�i. If ~H is a vague binary operation on A [ B, then

hA [ B; ~Hi 6v:s

hG; ~�i () ðiÞ l ~Hða; b; cÞ6 l~�ða; b; cÞ; 8a; b; c 2 A [ B; and

ðiiÞ A � B or B � A:

Proof. ()): hA; ~i 6v:s

hG; ~�i; hB; ~�i 6v:s

hG; ~�i and hA [ B; ~Hi 6v:s

hG; ~�i imply

hA,i 6 hG,�i,hB,•i 6 hG,�i and hA[B,wi 6 hG,�i, respectively. Therefore,we have A � B or B � A from [10]. On the other hand, hA [ B; ~Hi 6

v:s

hG; ~�ientails that l ~Hða; b; cÞ6l~�ða; b; cÞ for each a,b,c 2 A[B.

((): Without loss of the generality, we may assume that B �A. In this case,~H is a vague binary operation on A such that l ~Hðx; y; zÞ6 l~�ðx; y; zÞ for each

x,y,z 2 A. It is clear that hA; ~Hi is a vague semigroup. From hA; ~i 6v:shG; ~�i, we

have e 2 A and x�1 2 A for all x 2 A, i.e.,

hA [ B; ~Hi ¼ hA; ~Hi 6v:s

hG; ~�i: �

Corollary 25. Given a vague group hG; ~�i, let hA; ~i 6v:s

hG; ~�i and

hB; ~�i 6v:s

hG; ~�i. Then

hA [ B; ~�i 6v:s

hG; ~�i () A � B or B � A:


Proof. The desired result can be easily obtained from Proposition 24. h

Proposition 26. For a given vague group hG; ~�i and for ; 5 A�G, let ~ be a

vague binary operation on A. Then

hA; ~i6v:s

hG; ~�i()ðiÞ l~�ða;b�1;cÞ¼ 1) c2A; 8a;b2A; 8c2G; and

ðiiÞ l ~ða;b;cÞ6l~�ða;b;cÞ; 8a;b;c2A:

Proof. ()): (i) For a,b 2 A and c 2 G, let l~�ða; b�1; cÞ ¼ 1. If b 2 A, then b�1 2A. Then, by the equivalence (1), we obtain a � b�1 = c 2 A.

(ii) is obvious.

((): It is sufficient to show that hA; ~i is a vague group. It is easily seen that

hA; ~i is a vague semigroup. If a 2 A and e is an identity element of hG; ~�i, thenl~�ða; a�1; eÞ ¼ 1. Thus, e 2 A and l~�ðe; a1�; a�1Þ ¼ 1 imply a�1 2 A. Hence,

hA; ~i is a vague group, i.e.hA; ~i 6v:shG; ~�i. h

Corollary 27. Let hG; ~�i be a vague group. Then, for ; 5 A � G.

hA; ~�i 6v:s

hG; ~�i () ð8a; b 2 A; 8c 2 GÞðl~�ða; b�1; cÞ ¼ 1 ) c 2 AÞ:

Proof. A direct consequence of Proposition 26. h

Proposition 28. Let hG; ~�i be a vague group, ; 5 A � G and let ~ be a vaguebinary operation on A. Then

hA; ~i 6v:s

hG; ~�i $ðiÞ For each x 2 A; x�1 2 A; and

ðiiÞ l ~ða; b; cÞ6 l~�ða; b; cÞ; 8a; b; c 2 A:

Proof. ()): Obvious from Definition 21.

((): It is sufficient to show that e 2 A. For each x 2 A, there exists s 2 A

such that l ~ðx; x�1; sÞ ¼ 16 l~�ðx; x�1; sÞ. By the condition (F.2), we have

s = e 2 A. h

Corollary 29. Let hG; ~�i be a vague group and ; 5 A � G. If ~� is a vague binary

operation on A, then

hA; ~�i 6v:s

hG; ~�i () For each x 2 A; x�1 2 A:

Proof. Straightforward from Proposition 28. h


Corollary 30. Let hG; ~�i be a vague group and ~� be a vague binary operation on

G such that l~�ða; b; cÞ6 l~�ða; b; cÞ for all a,b, c 2 G. Let e be an identity element

of G. Then, hfeg; ~�i 6v:shG; ~�i and hG; ~�i 6

v:shG; ~�i.

Proof. Since e�1 = e 2 {e} and x�1 2 G for all x 2 G, and by Proposition 28,

we obtain that hfeg; ~�i 6v:shG; ~�i and hG; ~�i 6

v:shG; ~�i. h

Corollary 31. Let hG; ~�i be a vague group, and let hAj; ~�ji 6v:shG; ~�i for all j 2 J.

If ~H is a vague binary operation on \j2JAj such that

l ~Hðx; y; zÞ6^j2J

l~�jðx; y; zÞ; 8x; y; z 2 \j2JAj;

then h\j2JAj; ~Hi 6v:s

hAj; ~�ji.

Proof. If x 2 \j2JAj, then x 2 Aj for each j 2 J. Since hAj; ~�ji 6v:shG; ~�i, we have

x�1 2 Aj, i.e., x�1 2 \j 2 J Aj. Hence the required result immediately follows

from Proposition 28. h

Corollary 32. Let hG; ~�i be a vague group, A be a nonempty finite subset of G

and ~ be a vague binary operation on A. Then

hA; ~i 6v:s

hG; ~�i () l ~ða; b; cÞ6 l~�ða; b; cÞ; 8a; b; c 2 A:

Proof. ()): Obvious from Definition 21.

((): Since ~ is a vague binary operation on A, we have x � y 2 A for each

x,y 2 A. Thus, h A,�i 6 hG,�i, i.e., x�1 2 A for all x 2 A. From Proposition

28, we observe that hA; ~i 6v:shG; ~�i. h

Let hG; ~�i be a vague group, and hA; ~i a generalized vague subgroup ofhG; ~�i. If hA; ~i is not a crisp group, then the following proposition shows that

an infinite number of nontrivial generalized vague subgroups can be defined on

hA; ~i.

Proposition 33. Let hG; ~�i be a vague group, and hA; ~i 6v:shG; ~�i. For all

n 2 Nþ, let us define the fuzzy relation ~Hn : A� A,A by

l ~Hnðx; y; zÞ :¼1; l ~ðx; y; zÞ ¼ 1;l ~ðx;y;zÞ

n ; otherwise:

(

Then hA; ~Hni 6v:s

hA; ~i.

Proof. For each a,b,c,x,y,z 2 A, we have


l ~Hnða; b; cÞ ^ l ~Hnðx; y; zÞ ^ EA�Aðða; bÞ; ðx; yÞÞ

6l ~ða; b; cÞ ^ l ~ðx; y; zÞ ^ EG�Gðða; bÞ; ðx; yÞÞ

6EGðc; zÞ ¼ EAðc; zÞ

and l ~Hnða; b; a � bÞ ¼ 1. Thus, ~Hn is a vague binary operation on A. Using

Proposition 28, we obtain that hA; ~Hni 6v:s

hA; ~i. h

Proposition 34. Let hG; ~�i and hH ; ~Hi be two vague groups, and U : G! H a

vague homomorphism.

(i) If ~ is a vague binary operation on VKerU such that

l ~ðx; y; zÞ6 l~�ðx; y; zÞ;8x; y; z 2 VKerU;

then hVKerU; ~i 6v:s

hG; ~�i.(ii) If hA; ~i 6

v:s

hG; ~�i and ~� is a vague binary operation on U(A) such that

l~�ðx; y; zÞ6 l ~Hðx; y; zÞ for each x,y, z 2 U(A), then hUðAÞ; ~�i 6v:s

hH ; ~Hi.

(iii) If hB; ~�i 6v:s

hH ; ~Hi and ~ is a vague binary operation on U�1(B) such that

l ~ðx; y; zÞ6 l~�ðx; y; zÞ, for each x,y, z, z 2 U�1(B), then hU�1ðBÞ; ~i6

v:s

h G; ~�i.

Proof

(i) It is easy to see that

x 2 VKerU ) UðxÞ ¼ eH ) Uðx�1Þ ¼ eH ) x�1 2 VKerU:

Hence, hVKerU; ~i 6v:s

hG; ~�i follows from Proposition 28.(ii) For each x 2 U(A), there exists a 2 A such that U(a) = x. Because of

a�1 2 A, U(a�1) = x�1 2 U(A). From Proposition 28, we obtain that

hUðAÞ; ~�i 6v:s

hH ; ~Hi.(iii) For each x 2 U�1(B), there exists a 2 B such that U(x) = a. On the other

hand, since a�1 2 B, we have U(x�1) = (U(x))�1 = a�1 2 B. So

x�1 2 U�1(B). Then, from Proposition 28, hU�1ðBÞ; ~i 6v:s

hG; ~�i. h

It can be easily seen that Propositions 26, 28, Corollary 31 and Proposition 34

are a generalization of Theorems 4.2, 4.3, Corollary 4.4 and Theorem 4.11 in

[2], respectively.


5. Conclusions and remarks

In the present paper, some elementary properties of vague groups and some

properties of vague binary operations related with their associativity aspects

are obtained. Then concepts of external direct product of vague groups, vague

isomorphism and generalized vague subgroups are introduced, and various ba-sic properties of these concepts are investigated on the basis of the particular

integral, commutative cqm-lattice ([0, 1], 6 ,^). These concepts and their prop-

erties are explained with some examples.

Although the results in this paper are formulated on ([0,1], 6 ,^), it seems

that most of them can be restated for any t-norm instead of the minimum

t-norm ^, as in [9]. This topic is left to the readers for future investigations.

The concepts of vague ring, vague field and their algebraic properties are

studied on the basis of a general integral, commutative cqm-lattice (L, 6 ,w)in [6–8]. However, some of the other vague algebraic concepts for example va-

gue normal subgroup, vague subring, vague ideal (maximal and prime), vague

integer domain are introduced, and the validity of some relevant classical re-

sults in these settings are shown on the basis of the particular integral, commu-

tative cqm-lattice ([0,1], 6 ,^) in [12]. In forthcoming works, it is planned to

extend results, which are presented for the particular integral, commutative

cqm-lattice ([0,1], 6 ,^) in [12], to a general integral, commutative cqm-lattice

(L, 6 ,w).

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