Intuitionistic Fuzzy Topological BH-Algebras

: The intuitionistic fuzziﬁcation of a BH-algebra is considered and related results are investigated. The notion of equivalence relations on the family of all intuitionistic fuzzy BH-algebras of a BH-algebra is introduced, and then some properties are discussed. The concept of intuitionistic fuzzy topological BH-algebras is introduced, and some related results are obtained.


Introduction
Y. Imai and K. Iseki [8] introduced two classes of abstract algebras: BCK-algebras and BCIalgebras. It is known that the class of BCK-algebras is a proper subclass of the class of BCIalgebras. In [6,7] Q.P. Hu and X. Li introduced a wide class of abstract algebras: BCH-algebras. In 1996, Jun, Roh and Kim introduced the notion of BH-algebra, which is a generalization of BCH-algebras [9]. In 2001, Q. Zhang, E.H. Roh and Y.B. Jun studied the fuzzy theory in BHalgebras [14]. C.H. Park introduced the notion of an interval-valued fuzzy BH-algebra in a BHalgebra and investigate related properties [10]. The concept of a fuzzy set, which was introduced in [13], provides a natural framework for generalizing many of the concepts of general topology to what might be called fuzzy topological spaces. D. H. Foster [5] combined the structure of a fuzzy topological spaces with that of a fuzzy group, introduced by A. Rosenfeld [11], to formulate the elements of a theory of fuzzy topological groups. After the introduction of fuzzy sets by L. A. Zadeh [13], several researchers were conducted on the generalizations of the notion of fuzzy sets. The idea of intuitionistic fuzzy set was first published by K. T. Atanassov [1] , as a generalization of the notion of fuzzy sets. In this paper, using the Atanassovs idea, we establish the notion of intuitionistic fuzzy BH-algebras, equivalence relations on the family of all intuitionistic fuzzy BH-algebras, and intuitionistic fuzzy topological BH-algebras which are generalization of the notion of fuzzy topological BH-algebras. We investigate several properties, and show that the BH-homomorphic image and preimage of an intuitionistic fuzzy topological BH-algebra is an intuitionistic fuzzy topological BH-algebra. Definition 2.5. An intuitionistic fuzzy set (IFS for short) D in X is an object having the form where the functions µ D : X → [0, 1] and ν D : X → [0, 1] denote the degree of membership (namely µ D (x)) and the degree of nonmembership (namely ν D (x)) of each element x ∈ X to the set D, respectively, and For the sake of simplicity, we shall use the notation D = x, µ D , ν D instead of Let f be a mapping from a set X to a set Y. If 3 Intuitionistic fuzzy BH-algebras in X is called an intuitionistic fuzzy BH-algebra if it satisfies: Example 3.2. Consider a BH-algebra X = {0, a, b, c} with the following cayley table: is an intuitionistic fuzzy BH-algebra. and Proof. Let x ∈ X. Then Theorem 3.4. If {D i |i ∈ ∧} is an arbitrary family of intuitionistic fuzzy BH-algebras of X, then ∩D i is an intuitionistic fuzzy BH-algebra of X, where Proof. Let x, y ∈ X. Then Hence, is an intuitionistic fuzzy BH-algebra of X.
Theorem 3.5. If an IFS D = x, µ D , ν D in X is an intuitionistic fuzzy BH-algebra of X, then so is D, Proof. It is sufficient to show that µ D satisfies the second condition in Definition 3.1. Let x, y ∈ X. Then Hence D is an intuitionistic fuzzy BH-algebra of X.
Theorem 3.6. If an IFS D = x, µ D , ν D in X is an intuitionistic fuzzy BH-algebra of X, then the sets , and by applying Proposition 3.3 we conclude that ν D (x * y) = ν D (0) and hence x * y ∈ X ν .
Theorem 3.8. If an IFS D = x, µ D , ν D in X is an intuitionistic fuzzy BH-algebra of X, then the µ-level t-cut and ν-level t-cut of D are BH-subalgebras of X for every t ∈ [0, 1] such that t ∈ Im(µ D ) ∩ Im(ν D ), which are called a µ-level BH-subalgebra and a ν-level BH-subalgebra respectively.
Theorem 3.9. Let D = x, µ D , ν D be an IFS in X such that the sets U (µ D , t) and L(ν D , t) are BH-subalgebras of X. Then D = x, µ D , ν D is an intuitionistic fuzzy BH-algebra of X. Then . This is a contradiction, and therefore Taking This completes the proof.
Theorem 3.10. Any BH-subalgebra of X can be realized as both a µ-level BH-subalgebra and a ν-level BH-subalgebra of some intuitionistic fuzzy BH-algebra of X.
Proof. Let S be a BH-subalgebra of X and let µ D and ν D be fuzzy sets in X defined by for all x ∈ X where t and s are fixed numbers in (0, 1) such that t + s < 1.
If at least one of x and y does not belong to S, then at least one of µ D (x) and µ D (y) is equal to 0, and at least one of ν D (x) and ν D (y) is equal to 1. It follows that Hence D = x, µ D , ν D is an intuitionistic fuzzy BH-algebra of X. Obviously, This completes the proof. Theorem 3.11. Let α be a BH-homomorphism of a BH-algebra X into a BH-algebra Y and B an intuitionistic fuzzy BH-algebra of Y. Then α −1 (B) is an intuitionistic fuzzy BH-algebra of X.
Proof. For any x, y ∈ X, we have Hence α −1 (B) is an intuitionistic fuzzy BH-algebra in X.
Theorem 3.12. Let α be a BH-homomorphism of a BH -algebra X onto a BH-algebra Y. If D = x, µ D , ν D is an intuitionistic fuzzy BH-algebra of X, then α(D) = y, α sup µ D ), α inf (ν D ) is an intuitionistic fuzzy BH-algebra of Y.
Hence α t and β t are surjective.

Intuitionistic fuzzy topological BH-algebras
In [4], Coker generalized the concept of fuzzy topological space, first initiated by Chang [3], to the case of intuitionistic fuzzy sets as follows.

Definition 4.1 ([4]
). An intuitionistic fuzzy topology (IFT) on a non-empty set X is a family Φ of IFSs in X satisfying the following axioms: In this case the pair (X,Φ) is called an intuitionistic fuzzy topological space (IFTS for short) and any IFS in Φ is called an intuitionistic fuzzy open set (IFOS for short) in X.
This completes the proof. For any BH-algebra X and any element a ∈ X we use a r to denote the selfmap of X defined by a r (x) = x * a for all x ∈ X. Definition 4.6. Let X be BH-algebra, Φ an IFT on X and D an intuitionistic fuzzy BH-algebra with IIFT Φ D . Then D is called an intuitionistic fuzzy topological BH-algebra if for each a ∈ X the mapping a r : is relatively intuitionistic fuzzy continuous.
Theorem 4.7. Given BH-algebras X and Y, and a BH-homomorphism α : X → Y, let Φ and Ψ be the IFTs on X and Y, respectively such that Φ = α −1 (Ψ). If B is an intuitionistic fuzzy topological BH-algebra in Y, then α −1 (B) is an intuitionistic fuzzy topological BH-algebra in X.
Proof. Let a ∈ X and let U be an IFS in Φ α −1 (B). Since α is an intuitionistic fuzzy continuous mapping of (X, Φ) into (Y, Ψ), it follows from Proposition 4.5 that α is a relatively intuitionistic fuzzy continuous mapping of (α −1 (B), Φ α −1 (B)) into (B, Ψ B ). Note that there exists an IFS V in Ψ B such that α −1 (V ) = U. Then and Since B is an intuitionistic fuzzy topological BH-algebra in Y, the mapping is relatively intuitionistic fuzzy continuous for every b ∈ Y. Hence, This completes the proof.
Theorem 4.8. Given BH-algebras X and Y, and a BH-isomorphism α of X to Y, let Φ and Ψ be the IFTs on X and Y respectively such that α(Φ) = Ψ. If D is an intuitionistic fuzzy topological BH-algebra in X, then α(D) is an intuitionistic fuzzy topological BH-algebra in Y.
By hypothesis, the mapping  is an IFS in Ψ α(D) . This completes the proof.