t- Intuitionistic Fuzzy Subalgebra of BG-Algebras

The aim of this paper is to introduced the notion of t-intuitionistic fuzzy subalgebra and t-intuitionistic fuzzy normal subalgebra of BG-algebras. We state and prove some theorems in tintuitionistic fuzzy subalgebra and t-intuitionistic fuzzy normal subalgebra in BG-algebras. The homomorphic image and inverse image are investigated in both t-intuitionistic fuzzy subalgebra and normal subalgebras.


Introduction
In 1966, Imai and Iseki [6] introduced the two classes of abstract algebras, viz., BCK-algebras and BCI-algebras. It is known that the class of BCK-algebra is a proper subclass of the class of BCIalgebras. Neggers and Kim [8] introduced a new concept, called B-algebras, which are related to several classes of algebras such as BCI/BCK-algebras. Kim and Kim [7] introduced the notion of BG-algebra which is a generalization of B-algebra. The concept of intuitionistic fuzzy subset(IFS) was introduced by Atanassov [5] in 1983, which is a generalization of the notion of fuzzy sets. The concept of fuzzy subalgebras of BG-algebras was introduced by Ahn and Lee in [1]. The study of intuitionistic fuzzification of subalgebras and ideals of BG-algebras is done by Senapati et. al in [9]. The idea of t-intuitionistic fuzzy sets in fuzzy subgroups and fuzzy subrings is introduced by Sharma in [10,11]. Here in this paper, we introduced the notion of t-intuitionistic fuzzy sets in fuzzy subalgebra and fuzzy normal subalgebras of BG-algebras and study their properties.
For brevity, we also call X a BG-algebra. We can define a partial ordering " ≤ " on X by x ≤ y iff x * y = 0 ) ∀x ∈ X however equality hold when the map f is bijective.

Definition 0.4 ([2, 3]) An intuitionistic fuzzy set (IFS) A in a non empty set X is an object of the form
The numbers µ A (x) and ν A (x) denote respectively the degree of membership and the degree of non membership of the element x in the set A. For the sake of simplicity we shall use the symbol The class of IFSs on a universe X is denoted by IF S(X).

Definition 0.7 ([4]) For any IFS
Definition 0.8 ([9]) An intuitionistic fuzzy set A = (µ A , ν A ) of a BG-algebra X is said to be an intuitionistic fuzzy subalgebra of X if

t-Intuitionistic Fuzzy Subalgebra BG-algebra
Now onwards, let X denote a BG-algebra unless otherwise stated.

Definition 0.14 Let
is an intuitionistic fuzzy subalgebra BG-algebra X, then A is also t-intuitionistic fuzzy subalgebra of X.
is an intuitionistic fuzzy subalgebra BG-algebra X, therefore Hence A is also t-intuitionistic fuzzy subalgebra BG-algebra X.  The intuitionistic fuzzy subset hold. Hence A is t-intuitionistic fuzzy subalgebra of X.
is an intuitionistic fuzzy set of BG-algebra X and let t < min{p, 1−q}, where p = min{µ A (x)|x ∈ X} and q = max{ν A (x)|x ∈ X} then A is also t-intuitionistic fuzzy subalgebra BG-algebra X.
Proof. Since t < min{p, 1 − q} Hence A is t-intuitionistic fuzzy subalgebra of X.
Theorem 0.18 Any IF set of BG-algebra X can be realised as t-intuitionistic fuzzy subalgebra X.
Proof. It follows from Theorem 0.17 and Theorem 0.15.

Theorem 0.19
The intersection of two t-intuitionistic fuzzy subalgebra BG-algebra X is also a tintuitionistic fuzzy subalgebra of X.
Proof. Let x, y ∈ X. Then

Theorem 0.21 For every t-intuitionistic fuzzy subalgebra A t of X, the following properties hold
Theorem 0.22 If A be IF subalgebra of BG-algebra X, then A, ♢A and F α,β (A) are also t-intuitionistic fuzzy subalgebra of X.
Proof. Here A be IF subalgebra of BG-algebra X, By Theorem 0.15 A is also t-intuitionistic fuzzy subalgebra of X.
Hence by Eq n (1) and (3) Similarly we can show that 20 ATMath Volume 3 Theorem 0.23 Cartesian product of two t-intuitionistic fuzzy subalgebra of X is again a t-intuitionistic fuzzy subalgebra of X × X.
Theorem 0.25 If A = (µ A , ν A ) is an intuitionistic fuzzy normal subalgebra BG-algebra X, then A is also t-intuitionistic fuzzy normal subalgebra of X.
Proof. Since A = (µ A , ν A ) is an intuitionistic fuzzy normal subalgebra BG-algebra X, therefore Hence A is also t-intuitionistic fuzzy normal subalgebra BG-algebra X.
Remark 0.26 The converse of above Theorem is not true.
is an intuitionistic fuzzy set of BG-algebra X and let t < min{p, 1−q}, where p = min{µ A (x)|x ∈ X} and q = max{ν A (x)|x ∈ X} then A is also t-intuitionistic fuzzy normal subalgebra BG-algebra X.
Proof. Same as Theorem 0.17.
Theorem 0.28 The intersection of two t-intuitionistic fuzzy normal subalgebra BG-algebra X is also a t-intuitionistic fuzzy normal subalgebra of X.
Proof. Same as Theorem 0.19.
Theorem 0.29 If A be IF normal subalgebra of BG-algebra X,then A, ♢A and F α,β (A)are also t-intuitionistic fuzzy normal subalgebra of X.
Proof. Same as Theorem 0.22.
Theorem 0.30 Cartesian product of two t-intuitionistic fuzzy normal subalgebra of X is again a tintuitionistic fuzzy normal subalgebra of X × X.
Proof. Same as Theorem 0.23.
are also t-intuitionistic fuzzy normal subalgebra of X × X.

Homomorphism of t-intuitionistic fuzzy subalgebra BG-algebra
Definition 0.32 Let X and Y be two BG-algebras, then a mapping f : Theorem 0.33 Let f : X −→ Y be a homomorphism of BG-algebras, If A be a t-intuitionistic fuzzy subalgebra of Y , then f −1 (A)) is t-intuitionistic fuzzy subalgebra X.
Advanced Trends in Mathematics Vol. 3