t- Intuitionistic Fuzzy Subalgebra of BG-Algebras

S. R. BarbhuiyaDepartment of Mathematics,Srikishan Sarda College, Hailakandi,

Hailakandi-788151, Assam, Indiaa∗saidurbarbhuiya@gmail.com

Keywords: BG-algebra, Subalgebra, t-Intuitionistic Fuzzy Set, t-Intuiti onistic Fuzzy Subalgebra andt-Intuitionistic Fuzzy Normal Subalgebra.

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

Introduction

In 1966, Imai and Iseki [6] introduced the two classes of abstract algebras, viz., BCK-algebras andBCI-algebras. It is known that the class of BCK-algebra is a proper subclass of the class of BCI-algebras. Neggers and Kim [8] introduced a new concept, called B-algebras, which are related toseveral classes of algebras such as BCI/BCK-algebras. Kim and Kim [7] introduced the notion ofBG-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. Theconcept of fuzzy subalgebras of BG-algebras was introduced by Ahn and Lee in [1]. The study ofintuitionistic 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 Sharmain[10, 11]. Here in this paper, we introduced the notion of t-intuitionistic fuzzy sets in fuzzy subalgebraand fuzzy normal subalgebras of BG-algebras and study their properties.

Preliminaries

Definition 0.1 ([1]) A BG-algebra is a non-empty set X with a constant ′0′ and a binary operation′∗′ satisfying the following axioms:

(i) x ∗ x = 0,

(ii) x ∗ 0 = x,

(iii) (x ∗ y) ∗ (0 ∗ y) = x, ∀ x, y ∈ X.

For brevity, we also call X a BG-algebra. We can define a partial ordering ” ≤ ” on X by x ≤ y iffx ∗ y = 0

Definition 0.2 ([1]) A non-empty subset S of aBG-algebra X is called a subalgebra of X if x∗y ∈ S,for all x, y ∈ S.

Advanced Trends in Mathematics Vol. 3 (2015) pp 16-24 Online: 2015-06-01© (2015) SciPress Ltd., Switzerlanddoi:10.18052/www.scipress.com/ATMath.3.16

All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission ofSciPress Ltd., www.scipress.com. (ID: 49.137.99.223-13/07/15,16:07:20)

Definition 0.3 Let X and Y be two non empty sets and f : X −→ Y be a mapping. Let A andB be IFS’s of X and Y respectively. Then the image of A under the mapf is denoted by f(A) and is

defined by f(A)(y) = (µf(A)y, νf(A)y), where µf(A)(y) =

{∨{µA(x) : x ∈ f−1(y)}0 otherwise

νf(A)(y) ={∧{νA(x) : x ∈ f−1(y)}1 otherwisealso pre image of B under f is denoted by f−1(B) and is defined as f−1(B)(x) = (µf−1(B), νf−1(B) =

(µB(f(x)), νB(f(x)));∀x ∈ XRemark Note that µA(x) ≤ µf(A)(f(x)) and νA(x) ≥ νf(A)(f(x)) ∀x ∈ X however equality holdwhen 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 formA = {< x, µA(x), νA(x) > |x ∈ X} where µA : X → [0, 1] and νA : X → [0, 1] with the condition0 ≤ µA(x) + νA(x) ≤ 1, ∀x ∈ X .The numbers µA(x) and νA(x) denote respectively the degree ofmembership and the degree of non membership of the element x in the set A. For the sake of simplicitywe shall use the symbol A = (µA, νA) for the intuitionistic fuzzy set A = {< x, µA(x), νA(x) > |x ∈X}. The function πA(x) = 1 − µA(x) − νA(x) for all x ∈ X. is called the degree of uncertainty ofx ∈ A. The class of IFSs on a universe X is denoted by IFS(X).

Definition 0.5 ([2, 3]) If A = {< x, µA(x), νA(x) > |x ∈ X} and B = {< x, µB(x), νB(x) > |x ∈X} be any two IFS of a set X thenA ⊆ B iff for all x ∈ X,µA(x) ≤ µB(x) and νA(x) ≥ νB(x)A = B iff for all x ∈ X,µA(x) = µB(x) and νA(x) = νB(x)A ∩B = {< x, (µA ∩ µB)(x), (νA ∪ νB)(x) > |x ∈ X} where(µA ∩ µB)(x) = min{µA(x), µB(x)} and (νA ∪ νB)(x) = max{νA(x), νB(x)}A ∪B = {< x, (µA ∪ µB)(x), (νA ∩ νB)(x) > |x ∈ X} where(µA ∪ µB)(x) = max{µA(x), µB(x)} and (νA ∩ νB)(x) = min{νA(x), νB(x)}

Definition 0.6 ([4]) For any IFS A = {< x, µA(x), νA(x) > |x ∈ X} of X and α ∈ [01], theoperator � : IFS(X) → IFS(X),♢ : IFS(X) → IFS(X), Dα : IFS(X) → IFS(X) aredefined as(i) �(A) = {< x, µA(x), 1− µA(x) > |x ∈ X} is called necessity operator(ii) ♢(A) = {< x, 1− νA(x), νA(x) > |x ∈ X} is called possibility operator(iii)Dα(A) = {< x, µA(x)+απA(x), νA(x)+(1−α)πA(x) > |x ∈ X} is called α−Model operator.Clearly �(A) ⊆ A ⊆ ♢(A) and the equality hold, when A is a fuzzy set also D0(A) = �(A) andD1(A) = ♢(A). Therefore the α−Model operatorDα(A) is an extension of necessity operator�(A)and possibility operator ♢(A).

Definition 0.7 ([4]) For any IFS A = {< x, µA(x), νA(x) > |x ∈ X} of X and for any α, β ∈[01] such that α + β ≤ 1, the (α, β)− model operator Fα,β : IFS(X) → IFS(X) is defined asFα,β(A) = {< x, µA(x) + απA(x), νA(x) + βπA(x) > |x ∈ X}, where πA(x) = 1− µA(x)− νA(x)for all x ∈ X. Therefore we can writeFα,β(A) as Fα,β(A)(x) = (µFα,β(A)(x), νFα,β(A)(x))where µFα,β

(x) = µA(x) + απA(x) and νFα,β(A)(x)) = νA(x) + βπA(x).Clearly, F0,1(A) = �(A), F1,0(A) = ♢(A) and Fα,1−α(A) = Dα(A)

Definition 0.8 ([9]) An intuitionistic fuzzy set A = (µA, νA) of a BG-algebra X is said to be anintuitionistic fuzzy subalgebra of X if

(i)µA(x ∗ y) ≥ min{µA(x), µA(y)}(ii)νA(x ∗ y) ≤ max{νA(x), νA(y)} ∀x, y ∈ X.

Advanced Trends in Mathematics Vol. 3 17

Definition 0.9 ([7]) An IFS A of a BG-algebra X is said to be an IF normal subalgebra of X if(i) µA((x ∗ a) ∗ (y ∗ b)) ≥ min{µA(x ∗ y), µA(a ∗ b)},

(ii) νA((x ∗ a) ∗ (y ∗ b)) ≤ max{νA(x ∗ y), νA(a ∗ b)}, ∀ x, y ∈ X.

Definition 0.10 ([11]) Let A = (µA, νA) be an intuitionistic fuzzy set of BG-algebra X. Let t ∈ [0 1].then the intuitionistic fuzzy set At of X is called t-intuitionistic fuzzy subset(t-IFS) of X w.r.t A and isdefined by At = {< x, µAt(x), νAt(x) > |x ∈ X} =< µAt , νAt > where µAt(x) = min{µA(x), t}and νAt = max{νA(x), 1− t}∀x ∈ X

Remark 0.11 ([11]) LetAt =< µAt , νAt > andBt =< µBt , νBt > be two t-intuitionistic fuzzy subsetsof BG-algebra X, then(A ∩B)t = At ∩Bt

Remark 0.12 ([11]) Let f : X → Y be a mapping. Let A and B are two IFS of X and Y respectively,then(i)f−1(Bt) = (f−1(B))t (ii)f(At) = (f(A))t ∀t ∈ [0 1]

Definition 0.13 LetAt =< µAt , νAt > and Bt =< µBt , νBt > be two t-intuitionistic fuzzy subsets ofBG-algebra X Then their cartesian product At ×Bt =< µAt×Bt , µAt×Bt > is defined byµAt×Bt(x, y) = min{µAt(x), µAt(y)}νAt×Bt(x, y) = max{νAt(x), νAt(y)} ∀x, y ∈ X.

t-Intuitionistic Fuzzy Subalgebra BG-algebra

Now onwards, let X denote a BG-algebra unless otherwise stated.

Definition 0.14 Let A = (µA, νA) be an intuitionistic fuzzy set of BG-algebra X. Let t ∈ [0 1] then Ais called t-intuitionistic fuzzy subalgebra (t-IFSA) of X if At is IFSA of X i.e. if At satisfies followingconditions:

µAt(x ∗ y) ≥ min{µAt(x), µAt(y)}νAt(x ∗ y) ≤ max{νAt(x), νAt(y)}

Theorem 0.15 If A = (µA, νA) is an intuitionistic fuzzy subalgebra BG-algebra X, then A is alsot-intuitionistic fuzzy subalgebra of X.

Proof. Since A = (µA, νA) is an intuitionistic fuzzy subalgebra BG-algebra X, therefore

µA(x ∗ y) ≥ min{µA(x), µA(y)}νA(x ∗ y) ≤ max{νA(x), νA(y)}, ∀x, y ∈ X.

Now, µAt(x ∗ y) = min{µA(x ∗ y), t}≥ min{min{µA(x), µA(y)}, t}= min{min(µA(x), t),min(µA(y), t)}= min{µAt(x), µAt(y)}

⇒ µAt(x ∗ y) ≥ min{µAt(x), µAt(y)}Similarly we can show

νAt(x ∗ y) ≤ max{νAt(x), νAt(y)}

Hence A is also t-intuitionistic fuzzy subalgebra BG-algebra X.

18 Volume 3

Remark 0.16 The converse of above Theorem is not true.

Example 1. Consider a BG-algebra X = {0, 1, 2} with the following cayley table:

Table 1: Example of intuitionistic fuzzy BG-subalgebra.

* 0 1 20 0 1 21 1 0 12 2 2 0

The intuitionistic fuzzy subset A = {< x, µA(x), νA(x) > |x ∈ X} given by µA(0) = 0.4,µA(1) = 0.5, µA(2) = 0.3 and νA(0) = 0.5, νA(1) = 0.4, νA(2) = 0.6. SinceµA(0) = 0.4 ̸≥ min{µA(1), µA(1)}. Therefore A is not an intuitionistic fuzzy BG-subalgebra of X .Take t = 0.2. Then µA(x) > t for all x ∈ X and also νA(x) < 1− t for all x ∈ X.Therefore µAt(x ∗ y) ≥ min{µAt(x), µAt(y)} and νAt(x ∗ y) ≤ max{νAt(x), νAt(y)} for all x ∈ X.hold. Hence A is t-intuitionistic fuzzy subalgebra of X.

Theorem 0.17 IfA = (µA, νA) is an intuitionistic fuzzy set of BG-algebra X and let t < min{p, 1−q},where p = min{µA(x)|x ∈ X} and q = max{νA(x)|x ∈ X} then A is also t-intuitionistic fuzzysubalgebra BG-algebra X.

Proof. Since t < min{p, 1− q}

t < min{p, 1− q}⇒ p > t and 1− q > t

⇒ p > t and q < 1− t

⇒ min{µA(x)|x ∈ X} > t and max{νA(x)|x ∈ X} < 1− t

⇒ µA(x) > t, ∀ x ∈ X and νA(x) < 1− t,∀ x ∈ X

Therefore µAt(x ∗ y) ≥ min{µAt(x), µAt(y)} and νAt(x ∗ y) ≤ max{νAt(x), νAt(y)} for all x ∈ Xhold. 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 t-intuitionistic fuzzy subalgebra of X.

Advanced Trends in Mathematics Vol. 3 19

Proof. Let x, y ∈ X . Then

µ(A∩B)t(x ∗ y) = min{µ(A∩B)(x ∗ y), t}≥ min{min{µA(x ∗ y), µA(x ∗ y)}, t}= min{min(µA(x ∗ y), t),min(µB(x ∗ y), t)}= min{µAt(x ∗ y), µBt(x ∗ y)}≥ min{min{µAt(x), µAt(y)},min{µBt(x), µBt(y)}}= min{min{µAt(x), µBt(x)},min{µAt(y), µBt(y)}}= min{µ(A∩B)t(x), µ(A∩B)t(y)}

⇒ µ(A∩B)t(x ∗ y) ≥ min{µ(A∩B)t(x), µ(A∩B)t(y)}Similarly we can show that

ν(A∩B)t(x ∗ y) ≤ max{ν(A∩B)t(x), ν(A∩B)t(y)}

Theorem 0.20 The intersection of any number of t-intuitionistic fuzzy subalgebra BG-algebra X isalso a t-intuitionistic fuzzy subalgebra of X.

Theorem 0.21 For every t-intuitionistic fuzzy subalgebra At of X, the following properties hold(i) µAt(0) ≥ µAt(x)(ii) νAt(0) ≤ νAt(x), ∀x ∈ X.

Proof. We have µAt(0) = µAt(x ∗ x) ≥ min{µAt(x), µAt(x)} = µAt(x)and νAt(0) = νAt(x ∗ x) ≤ max{νAt(x), νAt(x)} = νAt(x)

Theorem 0.22 If A be IF subalgebra of BG-algebra X, then �A,♢A and Fα,β(A) are alsot-intuitionistic fuzzy subalgebra of X.

Proof. Here A be IF subalgebra of BG-algebra X, By Theorem 0.15 A is also t-intuitionistic fuzzysubalgebra of X.

µAt(x ∗ y) ≥ min{µAt(x), µAt(y)} (1)νAt(x ∗ y) ≤ max{νAt(x), νAt(y)} ∀x, y ∈ X. (2)

Now �At = {< x, µAt(x), 1− µAt(x)|x ∈ X} = {< x, µAt(x), µAt(x)|x ∈ X}♢At = {< x, 1− νAt(x), µAt(x)|x ∈ X} = {< x, νAt(x), µAt(x)|x ∈ X}

Now

µAt(x ∗ y) = 1− µAt(x ∗ y)≤ 1−min{µAt(x), µAt(y)} By(1)= max{1− µAt(x), 1− µAt(y)}= max{µAt(x), µAt(y)}

⇒ µAt(x ∗ y) ≤ max{µAt(x), µAt(y)} (3)

Hence byEqn (1) and (3)⇒ �At = {< x, µAt(x), µAt(x)|x ∈ X} is t-intuitionistic fuzzy subalgebraof X.Similarly we can show that

20 Volume 3

♢At = {< x, νAt(x), µAt(x)|x ∈ X} is t-intuitionistic fuzzy subalgebra of X.Again, we have Fα,β(A) =< µFα,β(A), νFα,β(A) > let x, y ∈ X, then Fα,β(x ∗ y) = (µFα,β(A)(x ∗y), νFα,β(A)(x ∗ y)) where µFα,β(A)(x ∗ y) = µA(x ∗ y)+απA(x ∗ y) and νFα,β(A)(x ∗ y) = νA(x ∗ y)+βπA(x ∗ y)

µFα,βAt(x ∗ y)= µAt(x ∗ y) + απAt(x ∗ y)= µAt(x ∗ y) + α(1− µAt(x ∗ y)− νAt(x ∗ y))≥ α+ (1− α)µAt(x ∗ y)− ανAt(x ∗ y)≥ α+ (1− α)min(µAt(x), µAt(y))− αmax(νAt(x), νAt(y)) By(1)≥ α{1−max(νAt(x), νAt(y))}+ (1− α)min(µAt(x), µAt(y))

≥ αmin(1− νAt(x), 1− νAt(y))}+ (1− α)min(µAt(x), µAt(y))

≥ min{α(1− νAt(x)) + (1− α)µAt(x), α(1− νAt(y)) + (1− α)µAt(y)}≥ min{µAt(x) + α(1− µAt(x)− νAt(x)), µAt(y) + α(1− µAt(y)− νAt(y))}≥ min{µFα,βAt(x), µFα,βAt(y)}

∴ µFα,βAt(x ∗ y) ≥ min{µFα,βAt(x), µFα,βAt(y)}Similarly we can prove thatνFα,βAt(x ∗ y) ≤ max{νFα,βAt(x), νFα,βAt(y)}

Hence Fα,β(A) is t-intuitionistic fuzzy subalgebra of X.

Theorem 0.23 Cartesian product of two t-intuitionistic fuzzy subalgebra of X is again a t-intuitionisticfuzzy subalgebra of X ×X .

Proof. LetAt =< µAt , νAt > and Bt =< µBt , νBt > be two t-intuitionistic fuzzy subalgebra of BG-algebra XThen their cartesian product At ×Bt =< µAt×Bt , µAt×Bt >, whereµAt×Bt(x, y) = min{µAt(x), µBt(y)}νAt×Bt(x, y) = max{νAt(x), νBt(y)} ∀x, y ∈ X.Also

µAt(x ∗ y) ≥ min{µAt(x), µAt(y)} (4)νAt(x ∗ y) ≤ max{νAt(x), νAt(y)} ∀x, y ∈ X. (5)

µAt×Bt((x1, y1) ∗ (x2, y2)) = µAt×Bt(x1 ∗ x2, y1 ∗ y2)= min{µAt(x1 ∗ x2), µBt(y1 ∗ y2)}≥ min{min{µAt(x1), µAt(x2)},min{µBt(y1), µBt(y2)}}= min{min{µAt(x1), µBt(y1)},min{µAt(x2), µBt(y2)}}= min{µAt×Bt((x1, y1), µAt×Bt((x2, y2)}

⇒ µAt×Bt((x1, y1) ∗ (x2, y2)) ≥ min{µAt×Bt((x1, y1), µAt×Bt((x2, y2)}Similarly we can show

νAt×Bt((x1, y1) ∗ (x2, y2)) ≤ max{νAt×Bt((x1, y1), νAt×Bt((x2, y2)}

Advanced Trends in Mathematics Vol. 3 21

Corollary 0.24 If At =< µAt , νAt > and Bt =< µBt , νBt > be two t-intuitionistic fuzzy subalgebraof BG-algebra X. Then�(At×Bt) ♢(At×Bt), Fα,β(A

t×Bt) are also t-intuitionistic fuzzy subalgebraof X ×X .

Theorem 0.25 If A = (µA, νA) is an intuitionistic fuzzy normal subalgebra BG-algebra X, then A isalso t-intuitionistic fuzzy normal subalgebra of X.

Proof. Since A = (µA, νA) is an intuitionistic fuzzy normal subalgebra BG-algebra X, therefore

(i)µA((x ∗ a) ∗ (y ∗ b)) ≥ min{µA(x ∗ y), µA(a ∗ b)}(ii)νA((x ∗ a) ∗ (y ∗ b)) ≤ max{νA(x ∗ y), νA(a ∗ b)}, ∀ x, y ∈ X.

Now, µAt((x ∗ a) ∗ (y ∗ b)) = min{µA((x ∗ a) ∗ (y ∗ b)), t}≥ min{min{µA(x ∗ y), µA(a ∗ b)}, t}= min{min(µA(x ∗ y), t),min(µA(a ∗ b), t)}= min{µAt(x ∗ y), µAt(a ∗ b)}

⇒ µAt((x ∗ a) ∗ (y ∗ b)) ≥ min{µAt(x ∗ y), µAt(a ∗ b)}Similarly we can show that

νAt((x ∗ a) ∗ (y ∗ b)) ≤ max{νAt(x ∗ y), νAt(a ∗ b)}

Hence A is also t-intuitionistic fuzzy normal subalgebra BG-algebra X.

Remark 0.26 The converse of above Theorem is not true.

Theorem 0.27 IfA = (µA, νA) is an intuitionistic fuzzy set of BG-algebra X and let t < min{p, 1−q},where p = min{µA(x)|x ∈ X} and q = max{νA(x)|x ∈ X} then A is also t-intuitionistic fuzzynormal 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 alsoa 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 Fα,β(A)are alsot-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 t-intuitionistic fuzzy normal subalgebra of X ×X .

Proof. Same as Theorem 0.23.

Corollary 0.31 If At =< µAt , νAt > and Bt =< µBt , νBt > be two t-intuitionistic fuzzy normalsubalgebra of BG-algebra X. Then �(At ×Bt), ♢(At ×Bt), Fα,β(A

t ×Bt) are also t-intuitionisticfuzzy normal subalgebra of X ×X .

Proof. Same as Corollary 0.24.

22 Volume 3

Homomorphism of t-intuitionistic fuzzy subalgebra BG-algebra

Definition 0.32 Let X and Y be two BG-algebras, then a mapping f : X −→ Y is said to behomomorphism if f(x ∗ y) = f(x) ∗ f(y), ∀x, y ∈ X .

Theorem 0.33 Let f : X −→ Y be a homomorphism of BG-algebras, If A be a t-intuitionistic fuzzysubalgebra of Y , then f−1(A)) is t-intuitionistic fuzzy subalgebra X.

Proof. A be a t-intuitionistic fuzzy subalgebra of Y. Let x, y ∈ X be any elements, then f−1(At)(x ∗y) = (µf−1(At)(x ∗ y), νf−1(At)(x ∗ y))

Now,µf−1(At)(x ∗ y)= µAtf(x ∗ y)= µAt [f(x) ∗ f(y)]≥ min{µAt(f(x)), µAt(f(y))} [Since A is t-IF subalgebra of Y]= min{µf−1At(x), µf−1At(y)}Therefore µf−1At(x ∗ y) ≥ min{µf−1At(x), µf−1At(y)}Similarly we can show thatνf−1(At)(x ∗ y) ≤ max{νf−1At(x), νf−1At(y)}

Hence, f−1(At) = (f−1(A))t is t-intuitionistic fuzzy subalgebra X.

Theorem 0.34 Let f : X −→ Y be a homomorphism of BG-algebras, If A be a t-intuitionistic fuzzynormal subalgebra of Y , then f−1(A) is t-intuitionistic fuzzy normal subalgebra X.

Theorem 0.35 Let f : X −→ Y be a onto homomorphism of BG-algebras, If A be t-intuitionisticfuzzy subalgebra X. Then f(A) is t-intuitionistic fuzzy subalgebra of Y.

Proof. Let y1, y2 ∈ Y Since f is onto, therefore there exists x1, x2 ∈ X such that f(x1) = y1, f(x2) =y2,f(A)(y1 ∗ y2) = (µf(A)(y1 ∗ y2), νf(A)(y1 ∗ y2)), Nowµf(A)(y1 ∗ y2) = µA(t) where f(t) = y1 ∗ y2 = f(x1) ∗ f(x2) = f(x1 ∗ x2)

µf(At)(y1 ∗ y2)= µ(f(A))t(y1 ∗ y2)= min{µf(A)(y1 ∗ y2), t}= min{µf(A)(f(x1) ∗ f(x2)), t}= min{µf(A)f(x1 ∗ x2), t}= min{µA(x1 ∗ x2), t}= µAt(x1 ∗ x2)

≥ min{µAt(x1), µAt(x2)}, forallx1, x2 ∈ X such that f(x1) = y1 and f(x1) = y1

= min{∨{µAt(x1)|f(x1) = y1},∨{µAt(x2)|f(x2) = y2}}= min{µf(At)(y1), µf(At)(y2)}Therefore µf(At)(y1 ∗ y2) ≥ min{µf(At)(y1), µf(At)(y2)}Similarly we can show thatνf(At)(y1 ∗ y2) ≤ max{νf(At)(y1), νf(At)(y2)}

Hence f(A) is t-intuitionistic fuzzy subalgebra of Y.

Advanced Trends in Mathematics Vol. 3 23

Theorem 0.36 Let f : X −→ Y be a onto homomorphism of BG-algebras, If A be t-intuitionisticfuzzy normal subalgebra X, then f(A) is t-intuitionistic fuzzy normal subalgebra of Y.

References

[1] S. S. Ahn and H. D. Lee, Fuzzy subalgebras of BG-algebras, Commun Korean Math.Soc 19(2)(2004) 243-251.

[2] K. T. Atanassov, Intutionistic fuzzy sets, Fuzzy sets and Systems.vol.1(1986),87-96

[3] K. T. Atanassov, More on intuitionistic Fuzzy Sets , Fuzzy sets and systems, 33(1) (1989), 37-45.

[4] K. T. Atanassov , On Intuitionistic Fuzzy Sets Theory, Published by Springer-Verlag Berlin Hei-delberg, 2012.

[5] K. T. Atanassov, Intuitionistic Fuzzy Sets, VII ITKR’s Session, Sofia,(Deposed in Central Sci. -Techn. Library of Bulg. Acad. of Sci., 1697/84) (June 1983)(in Bulg.)

[6] Y. Imai and K. Iseki, On Axiom systems of Propositional calculi XIV, Proc, Japan Academy, 42(1966)19-22.

[7] C. B. Kim and H.S. Kim, on BG-algebras, Demonstratio Mathematica, 41(3) 497-505.

[8] J. Neggers and H. S Kim, On B-algebras, Math. Vensik, 54 (2002), 21-29.

[9] T. Senapati, M. Bhowmik, M. Pal, Intuitionistic fuzzifications of ideals in BG-algebras, Mathe-matical Aeterna 2 (9) (2012) 761-778.

[10] P. K. Sharma, t-Intuitionistic Fuzzy Quotient Group, Advances in Fuzzy Mathematics, 7(1)(2012), 1-9.

[11] P. K. Sharma, t-Intuitionistic Fuzzy Subrings, IJMS, 11(3-4) (2012), 265-275.

[12] L. A. Zadeh, Fuzzy sets, Information and Control (1965), 338-353.

24 Volume 3

Volume 3 10.18052/www.scipress.com/ATMath.3 t-Intuitionistic Fuzzy Subalgebra of BG-Algebras 10.18052/www.scipress.com/ATMath.3.16