IMPLICATION-BASED T-FUZZY SUBGROUP OF A FINITE GROUP AND ITS PROPERTIES

Based on the definition of implication-based fuzzy subgroup of a finite group given by Yuan, implication-based Tfuzzy subgroup and implication-based Tnormal fuzzy subgroup of a finite group is defined. Some properties of them are proved in this paper. Also T− product, Ṫ− product and T ′ − product of these implication-based T-fuzzy subgroups of a finite group are defined and its properties are discussed. AMS Subject Classification: 03E72, 08A72, 20N25


Introduction
In 1965 the concept of fuzzy set was first introduced by Zadeh [1].Rosenfeld [2] and many others [3], [4], [5] have studied about the fuzzy normal subgroup.Anthony and Sherwood [6] redefined fuzzy subgroups using t-norm.Many including Sessa [7] studied about these T-fuzzy subgroups and proved many prop-erties.Yuan [8] defined implication-based fuzzy subgroup in 2003.In this paper we define the concept of implication-based T-fuzzy subgroup of a finite group and its properties.We also define the T − product, Ṫ − product and T ′ − product of these implication-based T-fuzzy subgroups of a finite group and proved some properties.

Preliminaries
Let X be an universe of discourse and (G, •) be a group.In fuzzy logic, truth value of fuzzy proposition α is denoted by [α].The fuzzy logical and the corresponding set theoretical notations used in this paper are: Then A is called a fuzzifying subgroup.

Definition 3. [8]
Let A be a fuzzy subset of a finite group G and λ ∈ (0, 1 ] is a fixed number.If for any x, y ∈ G Then A is called an implication-based fuzzy subgroup of G. Definition 4. [9] Let A be an implication-based fuzzy subgroup of G and f : G → G be a function defined on G. Then the implication-based fuzzy subgroup B of f (G) is defined by An homomorphic image or pre-image of an implicationbased fuzzy subgroup is an implication-based fuzzy subgroup provided in the former case the sup-property holds.=T (T (a, c), T (b, d)).Definition 6. [11] Given two t-norms T 1 and T 2 , T 1 is said to be stronger than T 2 , if T 1 (x, y) ≥ T 2 (x, y) ∀x, y ∈ [0, 1] and is written as and is written as Let (G, •) be a finite group with the identity element ′ e ′ , λ ∈ (0, 1 ] be a fixed number and let T : [0, 1] × [0, 1] → [0, 1] be a t − norm.
3. Implication-Based T-Fuzzy Subgroup and its Properties Definition 7. Let A be a fuzzy subset of G.For any x, y ∈ G if  Proof. where Since T is a continuous t-norm, for every ǫ > 0 there exists a δ > 0 such that Choose By the boundary conditions of the t-norm T, we have Theorem 6.Let A be an implication-based T-fuzzy subgroup of G and if there is a sequence {x n } in G such that lim n→∞ λ (T (( → (e ∈ A) By assumption, lim n→∞ λ (T (( Theorem 8. Let A be a fuzzy subset of G and T be a t-norm.If λ (e ∈ A) → 1 and λ (T ((x ∈ A), (y → (e(y −1 ) −1 ∈ A) This serves as the necessary and sufficient condition for a fuzzy subset A of G to be an implication-based T-fuzzy subgroup of G. Theorem 9. Let A be an implication-based T-fuzzy subgroup of G such that λ (a ∈ A) → 1. f a : G → G is called as right translation and is defined as f a (x) = xa and a f : G → G is called as left translation and is defined as Definition 12. Let G 1 and G 2 be two finite groups and G = G 1 × G 2 be the direct product group of G 1 and G 2 .Let A 1 and A 2 be two implicationbased T-fuzzy subgroups of G 1 and G 2 respectively and T ′ be a t-norm.Then the implication-based T ′ -fuzzy direct product of A 1 and Theorem 10.Let G 1 and G 2 be two finite groups and by the generalised associative law Let e = (e 1 , e 2 ) ∈ G where e 1 , e 2 are the identity elements of the group G 1 and G 2 respectively.
Theorem 13.Let A and B be two implication-based T-fuzzy normal subgroups of a group G. Let T ′ be a t-norm which dominates T. Then

Lemma 2 .
[10] Generalised Associative Law Let T :[0, 1] × [0, 1] → [0, 1] be a t-norm then T ((T (a, b)), (T (c, d))) = T ((T (a, c)), (T (b, d))) ∀a, b, c, d ∈ [0, 1] Proof.Let a, b, c, d ∈ [0, 1].ThenT ((T (a, b)), (T (c, d))) =T (a, T (b, T (c, d))) by (iii) and (iv) of definition of t-norm =T (a, T (T (b, c), d)) =T (a, T (T (c, b), d)) =T (a, T (c, T (b, d))) Example for implication-based T-fuzzy subgroup of a finite group.Consider the group G = {e, a, b, c} along with the binary operation ′ * ′ whose closure table is as follows.* e a b c e e a b c a a e c b b b c e a c c b a e For the fuzzy set A : G → [0, 1] defined by A(e) = 1, A(a) = .25,A(b) = .5,A(c) = .75with λ = .2and the implication operator is that of Lukasiewicz, with the t-norm defined by T (a, b) = ab we have T .75.1875.375.5625Then A is an implication-based T-fuzzy subgroup of G. Theorem 3. Let f be a homomorphism of the group G and B be an implication-based T-fuzzy subgroup of f

Theorem 4 .
Let A be an implication-based T-fuzzy subgroup of G and f be an homomorphism on G. Then B the image of A under f is also an implication-based T-fuzzy subgroup of f(G).

Definition 9 .
An implication-based T-fuzzy subgroup A of G is called an implication-based T-fuzzy normal subgroup if λ (xy ∈ A) → (yx ∈ A) ∀x, y ∈ G Theorem 5. Let A be an implication-based T-fuzzy normal subgroup of G such that λ (e ∈ A) → 1 then B = {x ∈ G/ λ (x ∈ A) → (e ∈ A)} is either empty or a normal subgroup of G. Proof.Let x, y ∈ B. Then λ (x ∈ A) → (e ∈ A) and λ (y ∈ A) → (e ∈ A).

Definition 10 .
Let A and B be two implication-based T-fuzzy subgroups of G. Then the implication-based T -fuzzy product of A and B denoted by [A • B] T is defined as λ (∃y, z{T ((y ∈ A), (z ∈ B))}; yz = x; y, z ∈ G) → (x ∈ A • B) ∀x ∈ G. Definition 11.Let A and B be two implication-based T-fuzzy subgroups of G. Then the implication-based Ṫ -fuzzy product of A and B denoted by [A • B] Ṫ is defined as

Theorem 11 .
Let G 1 and G 2 be two finite groups and G = G 1 × G 2 be the direct product of G 1 and G 2 .Let A and B be two implication-based Tfuzzy subgroups of G 1 and G 2 respectively.Then [A × B] T ′ is an implicationbased T-fuzzy subgroup of G provided T ′ dominates T. Moreover if A and B are implication-based T-fuzzy normal subgroup of G then [A × B] T ′ is an implication-based T-fuzzy normal subgroup of G. Proof.Let x, y ∈ G where x = (a 1 , b 1 ) and y = (a 2 , b 2