Intuitionistic Fuzzy Groups

INTRODUCTION

The theory of intuitionistic fuzzy set is expected to play an important role in modern mathematics in general as it represents a generalization of fuzzy set. The notion of intuitionistic fuzzy set was first defined by Atanassov (1986) as a generalization of Zadeh’s (1965) fuzzy set. After the concept of intuitionistic fuzzy set was introduced, several papers have been published by mathematicians to extend the classical mathematical concepts and fuzzy mathematical concepts to the case of intuitionistic fuzzy mathematics. The difficulty in such generalizations lies in how to pick out the rational generalization from the large number of available approaches. The study of fuzzy groups was first started with the introduction of the concept of fuzzy subgroups by Rosenfeld (1971). Anthony and Sherwood (1979) redefined fuzzy subgroups using the concept of triangular norm. In his remarkable paper Dib (1994) introduced a new approach to define fuzzy groups using his definition of fuzzy space which serves as the universal set in classical group theory. Dib (1994) remarked the absence of the fuzzy universal set and discussed some problems in Rosenfeld’s approach. In the case of intuitionistic fuzzy mathematics, there were some attempts to establish a significant and rational definition of intuitionistic fuzzy group. Zhan and Tan (2004) defined intuitionistic fuzzy subgroup as a generalization of Rosenfeld’s fuzzy subgroup. By starting with a given classical group they define intuitionistic fuzzy subgroup using the classical binary operation defined over the given classical group. In this study, to overcome the problems that will occur due to the absence of the concept of intuitionistic fuzzy universal set, we introduce the notion of intuitionistic fuzzy group based on the notion of intuitionistic fuzzy space and intuitionistic fuzzy function defined by Fathi and Salleh (2008a, b), which will serve as a universal set in the classical case.

PRELIMINARIES

Here, we will recall some of the fundamental concepts and definitions required in the sequel.

Let L = IxI, where I = [0, 1]. Define a partial order on L, in terms of the partial order on I, as follows: For every (r₁, r₂), (s₁, s₂)εL:

•	(r₁, r₂)≤(s₁, s₂), if r₁≤s₁, r₂≤s₂, whenever s₁≠s₂
•	(0, 0) = (s₁, s₂) whenever s₁ = 0 or s₂ = 0

Thus the cartesian product L = IxI is a distributive, not complemented lattice. The operation of infimum and supremum in L are given respectively by:

(r₁, r₂)Λ(s₁, s₂) = (r₁Λs₁, r₂Λs₂) and (r₁, r₂)ν(s₁, s₂) = (r₁νs₁, r₂νs₂).

Definition 1 (Atanassov, 1986)
Let X be a nonempty fixed set. An intuitionistic fuzzy set A is an object having the form:

where, the functions μ_A: X→I and v_A: X→I denote the degree of membership and the degree of nonmembership respectively of each element xεX to the set A and 0≤μ_A (x)+v_A(x)≤1 for all xεX.

Remark 1
The intuitionistic fuzzy set in X will be denoted by or simply where, and .

The support of the intuitionistic fuzzy set in X is the subset A of X defined by:

Definition 2
Let X be a nonempty set. An intuitionistic fuzzy space (simply IFS) denoted by (X, I , I) is the set of all ordered triples (x, I, I), where (x, I, I) = {(x, r, s): r, sεI with r+s≤1 and xεX} the ordered triplet (x, I, I) is called an intuitionistic fuzzy element of the intuitionistic fuzzy space (X, I, I) and the condition r, sεI with r+s≤1 will be referred to as the ’intuitionistic condition.

Therefore an intuitionistic fuzzy space is an (ordinary) set with ordered triples. In each triplet the first component indicates the (ordinary) element while the second and the third components indicate its set of possible membership and nonmembership values respectively.

Definition 3
Let U₀ be a given subset of X. An intuitionistic fuzzy subspace U of the IFS (X, I, I) is the collection of all ordered triples , where, xεU and are subsets of I such that u_x contains at least one element beside the zero element and contains at least one element beside the unit. If , then u_x = 0 and . The ordered triple will be called an intuitionistic fuzzy element of the intuitionistic fuzzy subspace U. The empty fuzzy subspace denoted by φ is defined to be:

Remark 2
For the sake of simplicity throughout this paper by saying an intuitionistic fuzzy space X we mean the intuitionistic fuzzy space (X, I, I).

Let X be an intuitionistic fuzzy space and A be an intuitionistic fuzzy subset of X. The fuzzy subset A induces the following intuitionistic fuzzy subspaces:

The lower-upper intuitionistic fuzzy subspace induced by A:

The upper-lower intuitionistic fuzzy subspace induced by A:

The finite intuitionistic fuzzy subspace induced by A is denoted by:

Definition 4
Let X and Y be any two intuitionistic fuzzy spaces. The intuitionistic fuzzy cartesian product denoted by (X, I, I)(Y, I, I) is defined as follows:

Definition 5
An intuitionistic fuzzy relation ρ from an IFS X to an IFS Y is a subset of the intuitionistic fuzzy cartesian product (X, I, I)(Y, I, I). An intuitionistic fuzzy relation from an IFS X into itself is called an intuitionistic fuzzy relation in the IFS X.

From the above definition we note that an intuitionistic fuzzy relation from an IFS X to an IFS Y is simply a collection of intuitionistic fuzzy subsets of (X, I, I)(Y, I, I), also the intuitionistic fuzzy cartesian product (X, I, I)(Y, I, I) is itself an intuitionistic fuzzy relation.

Definition 6
An intuitionistic fuzzy function between two intuitionistic fuzzy spaces X and Y is an intuitionistic fuzzy relation F from the X to Y satisfying the following conditions:

•	For every xεX with r, r, sεI, there exists a unique element yεY with w, zεI; such that ((x, y), (r, w), (s, z)) for some AεF
•	If ((x, y),(r₁, w₁), (s₁, z₁))εAε F and ((x, y’), (r₂, w₂), (s₂, z₂))εBε F then y = y’
•	If ((x, y),(r₁, w₁), (s₁, z₁))εAε F, and ((x, y’), (r₂, w₂), (s₂, z₂))εBε F then (r₁>r₂) implies (w₁>w₂) and (s₁>s₂) implies (z₁>z₂)
•	If ((x, y),(r₁, w₁), (s₁, z₁))εAε F, then r = 0 implies w = 0, s = 1 implies z = 0 and r = 1 implies w = 1, s = 0 implies z = 1

Thus conditions 1 and 2 imply that there exists a unique (ordinary) function from X to Y, namely F: X→Y and that for every xεX there exists unique (ordinary) functions from I to I, namely . On the other hand conditions 3 and 4 are respectively equivalent to the following conditions:

•	are nondecreasing on I.
•

That is, an intuitionistic fuzzy function between two intuitionistic fuzzy spaces X and Y is a function F from X to Y characterized by the ordered triple:

where, F(x) is a function from x to Y and are family of functions from I to I satisfying the conditions (I) and (ii) such that the image of any intuitionistic fuzzy subset A of the IFS X under F is the intuitionistic fuzzy subset F^(A) of the IFS Y defined by:

We will call the functions the comembership functions and the cononmembership functions, respectively. The intuitionistic fuzzy function F will be denoted by:

INTUITIONISTIC FUZZY GROUPS

Here, we introduce the concept of intuitionistic fuzzy group and study some of its properties. First we start by defining intuitionistic fuzzy binary operation on a given IFS.

Definition 7
An intuitionistic fuzzy binary operation F on an IFS (X, I, I) is an intuitionistic fuzzy function F: XxX→X with comembership functions f_xy and cononmembership functions satisfying:

•	iff r≠0 and s≠0 and iff w≠1 and z≠1
•	are onto. That is, and

Thus for any two intuitionistic fuzzy elements (x, I, I), (y, I, I) of the IFS X and any intuitionistic fuzzy binary operation F = defined on an IFS X. The action of the intuitionistic fuzzy binary operation F = over the IFS X is given by:

An intuitionistic fuzzy binary operation is said to be uniform if the associated comembership and cononmembership functions are identical. That is, if for all x, y εX. A left semiuniform (right semiuniform) fuzzy binary operation is an intuitionistic fuzzy binary operation having identical comembership functions (cononmembership functions).

Definition 8
An intuitionistic fuzzy groupoid, denoted by ((X, I, I), F), is an IFS (X, I, I) together with an intuitionistic fuzzy binary operation F defined over it. A uniform (left semiuniform, right semiuniform) intuitionistic fuzzy groupoid is an intuitionistic fuzzy groupoid with uniform (left semiuniform, right semiuniform) intuitionistic fuzzy binary operation.

Theorem 1
Associated to each intuitionistic fuzzy groupoid ((X, I, I), F) where, are two fuzzy groupoids, namely ((X, I), F) and where, F (F, f_xy) and which are isomorphic to the intuitionistic fuzzy groupoid ((X, I, I), F) by the correspondence (x, I, I)↔(x, I).

To each intuitionistic fuzzy groupoid ((X, I, I), F) there is an associated (ordinary) groupoid (X, F) which is isomorphic to the intuitionistic fuzzy groupoid by the correspondence (x, I, I)↔x.

Definition 9
The ordered pair (U; F) is said to be an intuitionistic fuzzy subgroupoid of the intuitionistic fuzzy groupoid ((X, I, I), F) iff U is an intuitionistic fuzzy subspace of the intuitionistic fuzzy space X and U is closed under the intuitionistic binary operation F.

Definition 10
An intuitionistic fuzzy semigroup is an intuitionistic fuzzy groupied that is associative. An intuitionistic fuzzy monoid is an intuitionistic fuzzy semigroup that admits an identity.

After defining the concepts of intuitionistic fuzzy groupoid, intuitionistic fuzzy semigroup and intuitionistic fuzzy monoid we introduce now the notion of intuitionistic fuzzy group.

Definition 11
An intuitionistic fuzzy group is an intuitionistic fuzzy monoid in which each intuitionistic fuzzy element has an inverse. That is, an intuitionistic fuzzy groupoid ((G, I, I), F) is an intuitionistic fuzzy group iff the following conditions are satisfied:

•	For any choice of (associativity)
•	There exists an intuitionistic fuzzy element (e, I, I) ε(C, I, I) such that for all (X, I, I) in ((G, I, I), F) : (e, I, I)F(x, I, I) = (x, I, I)F(e, I, I) = (x, I, I) (existence of an identity)
•	For every intuitionistic fuzzy element (x, I, I) in ((G, I, I), F) there exists an intuitionistic fuzzy element (x^-1, I, I) in ((G, I, I), F) such that: (x, I, I)F(x^-1, I, I) = (x^-1, I, I) = (e, I, I) (existence of an inverse)

An intuitionistic fuzzy group ((G, I, I), F) is called an abelian (commutative) intuitionistic fuzzy group if and only if for all (x, I, I), (y, I, I)ε((G, I, I), F)(x, I, I)F(y, I, I) = (y, I, I)F(z, I, I).

By the order of an intuitionistic fuzzy group we mean the number of intuitionistic fuzzy elements in the intuitionistic fuzzy group. An intuitionistic fuzzy group of infinite order is an infinite intuitionistic fuzzy group.

Remark 3
Throughout this study by saying an intuitionistic fuzzy group we mean an intuitionistic fuzzy group based on intuitionistic fuzzy space and by saying a classical intuitionistic fuzzy group we mean an intuitionistic fuzzy subgroup based on Rosenfeld’s approach.

Theorem 2
Associated to each intuitionistic fuzzy group ((G, I, I), F) where, two fuzzy groups, namely where, and which are isomorphic to the intuitionistic fuzzy group ((G, I, I), F) by the correspondence (x, I, I)↔(x, I).

To each intuitionistic fuzzy group ((G, I, I), F) there is an associated (ordinary) group (G, F) which is isomorphic to the intuitionistic fuzzy group by the correspondence (x I, I)↔x.

As a result of Theorem 2 (which we will refer to as the associativity theorem) a sufficient and necessary condition for intuitionistic fuzzy group is given in the following corollary:

Corollary 1
Let (X, I, I) be an intuitionistic fuzzy space and let be an intuitionistic fuzzy binary operation defined over (X, I, I). ((X, I, I), F) defines an intuitionistic fuzzy group iff ((X, I, F)) and are both fuzzy groups.

Example 1
Consider the set G = {a}. Define the intuitionistic fuzzy binary operation over the intuitionistic fuzzy space (G, I, I) such that: F (a, a) = a and .

Thus, the intuitionistic fuzzy space (G, I, I) together with F define a (trivial) intuitionistic fuzzy group ((G, I, I), F).

Consider the set Z₃ = {0, 1, 2}. Define the intuitionistic fuzzy binary operation over the intuitionistic fuzzy space (Z₃, I, I) as follows: F (x, y) = x+₃y, where, +3 refers to addition modulo 3 and . Then ((Z₃, I, I), F) is an intuitionistic fuzzy group.

The next theorem follows directly from the definition of intuitionistic fuzzy group and the associativity theorem.

Theorem 3
For any intuitionistic fuzzy group ((G, I, I), F), the following are true:

•	The intuitionistic fuzzy identity element is unique
•	The inverse of each intuitionistic fuzzy element (x, I, I)ε((G, I, I), F) is unique
•	((x^-1)^-1, I. I) = (x, I, I)
•	For all (x, I, I), (y, I, I)ε((G, I, I), F), ((x. I, I)F(y, I, I))^-1 = (y^-1, I, I)F(x^-1, I, I)
•	For all (x, I, I)F(y, I, I), (z, I, I)ε((G, I, I), F)

If , then

if , then

Definition 12
Let S be an intuitionistic fuzzy subspace of the IFS (G, I, I). The ordered pair (S; F) is an intuitionistic fuzzy subgroup of the IFG ((G, I, I), F), denoted by (S; F)≤((G, I, I), F), if (S; F) defines an IFG under the intuitionistic fuzzy binary operation F.

Obviously, if (S; F) is an intuitionistic fuzzy subgroup of ((G, I, I), F) and (T; F) is an intuitionistic fuzzy subgroup of (S; F), then (T; F) is an intuitionistic fuzzy subgroup of (G, I, I), F). Also if (G, I, I), F) is an IFG with an intuitionistic fuzzy identity (e, I, I) then both {(e, I, I), F} and ((G, I, I), F) are (trivial) intuitionistic fuzzy subgroups of ((G, I, I), F).

Theorem 4
Let be an intuitionistic fuzzy subspace of the intuitionistic fuzzy space (G, I, I). Then (S; F) is an intuitionistic fuzzy subgroup of the IFG ((G, I, I), F) iff:

•	(S_o; F) is an (ordinary) subgroup of the group (G, F)
•	and , for all x, yεS_o

Proof
If (1) and (2) are satisfied, then:

•	The intuitionistic fuzzy subspace S is closed under F: Let be in S then

•	(S; F) satisfies the conditions of intuitionistic fuzzy group:
•	Let and be in S then:

•	Since (S_o; F) is an (ordinary) subgroup of the group (G, F) then S_o contains the identity e. That is, thus:

Similarly we can show that .

•	Again, since (S_o, F) is an (ordinary) subgroup of the group ((G, I, I), F) then S_o contains the inverse element x^-1 for each xεs, that is, for each, then:

Similarly we can show that .

By (I) and (ii) we conclude that (S; F) is an intuitionistic fuzzy subgroup of ((G, I, I), F).

Conversely if (S; F) is an intuitionistic fuzzy subgroup of ((G, I, I), F) then (1) holds by the associativity theorem. Also being onto over the partial ordered sub-lattice s_xxs_y of the vector lattice IxI.

Example 2

•

Let ((G, I, I), F) be defined as in Example 2(1). Consider the fuzzy subspace such that 0<α, β<1. Then (S, F) defines an intuitionistic fuzzy subgroup of ((G, I, I), F). If we consider S’ = {(α, [0, γ], [δ, 1])} such that 0<γ, δ<1 and α≠γ, β≠δ then (S’, F) defines an intuitionistic fuzzy subgroup of ((G, I, I), F), where, S≠S’. That is, a trivial intuitionistic fuzzy group can have more than one intuitionistic fuzzy subgroup, which is different from the case of ordinary groups, since an ordinary trivial group can only have one subgroup, namely the group itself.

•

Let (Z₃, I, I), F) be defined as in Example 2(2). Consider the fuzzy subspace such that . Then (Z, F) is not an intuitionistic fuzzy subgroup of ((G, I, I), F), since Z is not closed under F. That is:

(0, [0, α], [β, 1])F(1, [0, γ], [δ, 1]) = (1, [0, α·γ], [β·δ, 1])≠Z

If α, γ = 1 and β, δ = 0, then Z {(0, I, I), (1, I, I)} together with F defines an intuitionistic fuzzy subgroup of ((G, I, I), F).

INTUITIONISTIC FUZZY SUBGROUPS INDUCED BY INTUITIONISTIC
FUZZY SUBSETS

Here, we introduce intuitionistic fuzzy subgroups induced by intuitionistic fuzzy subsets and then we obtain a relationship between intuitionistic fuzzy subgroups and classical fuzzy subgroups.

Let A be an intuitionistic fuzzy subset of the set G and let H_lu (A), H_ul (A) and H_o (A) be intuitionistic fuzzy subspaces induced by the fuzzy subset A. For these intuitionistic fuzzy spaces we can re-state Theorem 3.13 in the following manner.

Theorem 5
(H_lu (A), F), (H_ul (A), F) and (H_o, (A), F) are intuitionistic fuzzy subgroups of ((G, I, I), F) iff

•	xFyεA_o for all x, yεA_o
•	f_xy (A(x), A(y)) = A (xFy) and

Definition 13
An intuitionistic fuzzy subset A of G induces intuitionistic fuzzy subgroups of ((G, I, I), F) iff (H_lu (A), F), (H_ul (A), F) and (H_o (A), F) are intuitionistic fuzzy subgroups.

Let ((G, I, I), F) with be a uniform intuitionistic fuzzy group with having the t-norm properties, then we have the following theorem:

Theorem 6

•	Every intuitionistic fuzzy subset A of G which induces intuitionistic fuzzy subgroups is a classical fuzzy subgroup of (G, F).
•	If (S, F) is an ordinary subgroup of the group (G, F) then every intuitionistic fuzzy subset A of G for which A_o = S induces an intuitionistic fuzzy subgroup of ((G, I, I), P) where, with P = F and are suitable comembership and cononmembership functions, respectively.

Proof

•	If the intuitionistic fuzzy subset A induces intuitionistic fuzzy subgroups of ((G, I, I), P) then by Theorem 4.1 we have and , for all A(x)≠0≠A(y) and

That is, if the intuitionistic fuzzy subset A induces intuitionistic fuzzy subgroups of ((G, I, I), F) then it satisfies the inequalities and , for all x, yεG. Therefore, A is a classical intuitionistic fuzzy subgroup.

•	Let (S, F) be an ordinary subgroup of the group (G, F) and let A be an intuitionistic fuzzy subset of G for which A_o = S and let be given t-norms. Define the intuitionistic fuzzy group ((G, I, I), F) as follows:

where, P = F and such that:

If , then for all tεI. If , then:

and if , then for all kεI. If , then:

It is clear that are continuous comembership and cononmembership functions, respectively. Moreover iff r₁ = 0 or r₂ = 0 and iff s₁ = 1 or s₂ = 0. Hence P is an intuitionistic fuzzy binary operation on G.

Now, based on the property of the given t-norms and the construction of we notice that whenever both A(x), A(y) have nonzero values and whenever both are not equal to one. That is:

Thus, by Theorem 4.1 and the assumption that (S, F) is an ordinary subgroup A induces an intuitionistic fuzzy subgroup of the intuitionistic fuzzy group ((G, I, I), P).

Corollary 2
Every classical intuitionistic fuzzy subgroup A of (G, F) induces intuitionistic fuzzy subgroups relative to some intuitionistic fuzzy group (G, P).

CONCLUSION

In this study, we have generalized the study initiated in (Dib, 1994) about fuzzy groups to the context of intuitionistic fuzzy groups. In the absence of the intuitionistic fuzzy universal set, formulation of the intrinsic definition for an intuitionistic fuzzy subgroup is not evident. In this paper we define the notion of an intuitionistic fuzzy group using the notion of an intuitionistic fuzzy space. The use of intuitionistic fuzzy space as a universal set corrects the deviation in the definition of intuitionistic fuzzy subgroups. This concept can be considered as a new formulation of the classical theory of intuitionistic fuzzy groups.

ACKNOWLEDGMENT

The financial support received from University Kebangsaan Malaysia, Faculty of Science and Technology under grant No. (UKM-ST-06-FRGS0009-2007) is gratefully acknowledged.

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Asian Journal of Algebra

Year: 2009 | Volume: 2 | Issue: 1 | Page No.: 1-10
DOI: 10.3923/aja.2009.1.10

M. Fathi and A.R. Salleh

How to cite this article

M. Fathi and A.R. Salleh, 2009. Intuitionistic Fuzzy Groups. Asian Journal of Algebra, 2: 1-10.

Keywords: Intuitionistic fuzzy space, intuitionistic fuzzy binary operation and intuitionistic fuzzy group

REFERENCES