New category of the Fuzzy d-algebras

ABSTRACT The aim of this work is to introduce a new category of Fuzzy d-algebra called Γ-Fuzzy d-algebra. This notion is more useful than Fuzzy d-algebra because in the old case the degree of membership of any in the Fuzzy set does not depend on the structure of d-algebra X. Therefore, in this work, we introduced a new construction of Fuzzy d-algebra that is depended on structure of d-algebra. Also, we show that for any d-algebra there is a Γ-Fuzzy d-algebra. Further, this new construction helps us to find the interesting relation between Γ-Fuzzy d-algebra and edge d-algebra in this work. For any d-algebra, we determined unique Γ-Fuzzy topological space using Γ-Fuzzy d-algebra. For any , a new class of d-algebra is introduced, where is the set of all equivalence classes . Finally, a new class of intuitionistic fuzzy set is called intuitionistic Γ-Fuzzy d-algebra is investigated and discussed.


Introduction
In 1966, two classes of abstract algebras were introducedthe first one was BCK-algebra, which was introduced by Imai and Iseki [1] and other was BCI-algebra, which was introduced by Iseki [2]. After that, a wide class of abstract algebras (BCH-algebras) were introduced by Hu [3] in 1983 and Li [4] in 1985. Further, they showed that the classes of BCK-algebras were proper subclasses of the classes of BCI-algebras and the classes of BCI-algebras were proper subclasses of the classes of BCH-algebras. Next, some new classes of algebras are given ( [5][6][7][8]). A Fuzzy set was a class of objects with a continuum of grades of membership was a concept proposed by Zadeh [9] in 1965. After the introduction of Fuzzy topology by Chang [10] in 1968, there have been several generalizations of notions of Fuzzy set and Fuzzy topology.
By adding the degree of non-membership to fuzzy set, Atanassov [11] proposed intuitionistic fuzzy set (IFS) in 1986 which looks more accurate to uncertainty quantification and provides the opportunity to precisely model the problem based on the existing knowledge and observations. In recent years, some interesting studies on these sets have been discussed by Shuker [12][13][14][15] and others.
In 1993, Jun [16] combined the structure of Fuzzy topological spaces with that of a Fuzzy BCK-algebras to formulate the elements of a theory of Fuzzy topological BCK-algebras. In 1999, the concept of d-algebra, which is another generalization of BCK-algebras, is introduced by Neggers and Kim [17]. Also, the notion of d-ideal in d-algebra is discussed by Jun, Neggers and Kim [18]. After that, they introduced the notions of fuzzy d-subalgebra, fuzzy d-ideal, fuzzy d # -ideal, fuzzy d*-ideal, fuzzy B-algebras, fuzzy BCI-algebras* and the relations among them are shown, see [19][20][21].
In this paper, new construction of Fuzzy d-algebra is called Γ-Fuzzy d-algebra, this notation is more useful than Fuzzy d-algebra because the degree of membership of any x [ X in the Fuzzy set does not depend on the structure of d-algebra X. Therefore, in this work, we introduced a new construction of Fuzzy d-algebra that is dependent on the structure of d-algebra. Moreover, in this paper, we show that for any d-algebra there is a Γ-Fuzzy d-algebra. Further, the interesting relation between Γ-Fuzzy d-algebra and edge dalgebra are shown. For any d-algebra, we determined unique Γ-Fuzzy topological space (X, G f ) using Γ-Fuzzy d-algebra. Also, in this work, for any f [ X a new class of d-algebra (C(X), 4, (X, Finally, a new class of IFS called intuitionistic Γ-Fuzzy dalgebra is investigated and its applications are discussed.

Definitions and notations
The following definitions have been used to obtain the results and properties developed in this paper.
Definition 2.1: [17] A d-algebra is a nonempty set X with a constant 0 and a binary operation* satisfying the following axioms: (1) x * x = 0 (2) 0 * x = 0 (3) x * y = 0 and y * x = 0 imply that y = x for all x, y in X. For a set X, we define a Fuzzy set in X to be a function μ: X [0, 1]. Here, μ(x) "represents the degree of membership of x in the Fuzzy set A".
[Note]: Any subset A of a set X can be identified with its characteristic function x A : X → {0, 1} defined by Definition 2.6: [10] A Fuzzy topology on a set X is a collection δ of Fuzzy sets in X satisfying: Definition 2.7: [11] An IFS A over the universe X can be defined as follows The values m A (x) and v A (x) represent the degree of membership and non-membership of x to A, respectively. Remark 2.9: [11] p A (x)expresses the lack of knowledge of whether x belongs to IFS A or not. For example, let A be an IFS with m A (x) = 0.5 and n A (x) = 0.3 ⇒ p A (x) = 1 − (0.5 + 0.3) = 0.2. It can be interpreted as "the degree that the object x belongs to IFS A is 0.5, the degree that the object x does not belong to IFS A is 0.3 and the degree of hesitancy is 0.2".

Note on fuzzy d-algebras
In this section, we will explain that in the fuzzy dalgebra X the degree of membership of any x [ X in the fuzzy set does not depend on structure of dalgebra X.
Example 3.2: Let X = {0, a, b, c} be a set with binary operation ( * ) defined in Table 1. Table 1, we consider that the values l and r have already existed and hence these values do not depend on the structure of d-algebra X. On the other hand, let (X, * , 0) be dalgebra with Table 1 Table 2. Then (X, * , 0) is not d-algebra. However, for Table 2, we consider that That means there is no relation between Fuzzy set and the structure of d-algebra. This fact is also true for the structure of edge d-algebra. Therefore, in this paper, we will introduce new constructions of Fuzzy dalgebra that is dependent on the structure of dalgebra is called Γ-Fuzzy d-algebra and explain its application on edge d-algebra.

Γ-Fuzzy d-algebras
In this section, we will introduce a new concept called Γ-Fuzzy d-algebra and show that for any d-algebra Table 2. (X, * , 0) is not d-algebra.
there is a Γ-Fuzzy d-algebra. Further, this new construction helps us to find the interesting relation between Γ-Fuzzy d-algebra and edge d-algebra.
Then Γ is said to be Γ-Fuzzy d-algebra.

Γ-Fuzzy d-algebras and their applications
Suppose that m is Γ-Fuzzy d-algebra of X. Then we will consider a new method to check that if X is edge or not edge. In this case, for any (X, * , 0) d-algebra, we can use the form of its Γ-Fuzzy (see Theorem (5.1)). Also, for any (X, * , 0) d-algebra, a new class in Fuzzy topological spaces called Γ-Fuzzy topological space is investigated. Moreover, we will consider that X is edge d-algebra if and only if its Γ-Fuzzy topological space is Γ-Fuzzy indiscrete topological space. Further, for any (X, * , 0) d-algebra, a new class of IFS is called intuitionistic Γ-Fuzzy d-algebra is investigated and its applications are discussed.
Example 5.7: Let (X, * , f ) be d-algebra where X = {f , a, b, c} with the binary operation * defined by Table 5. Table 6.
Then (X, ⊗ x k , x k ) is d-algebra and its Γ-Fuzzy d-algebra is denoted by m ⊗ x k .
Remarks 5.14: (1) Since ≈ is an equivalence relation on j(X), then the equivalence classes form a partition of j(X).
(2) For any nonempty set X = {x 1 , x 2 , x 3 , . . . }, we consider that (X, Then m ⊗ x k is an absolutely Γ-Fuzzy d-algebra and hence (X, ⊗ x k , x k ) is edge d-algebra.
Definition 5.18: Define C(X) as the set of all equivalence classes (X, * , f ) l , and equip C(X) with the following binary operation 4:C(X) × C(X) C(X) that is defined as follows: Proof: We need to prove the following: Then then this implies that (l = 1 X or l = h) and (h = 1 X or h = l). Hence, there are four cases that cover all probabilities, which are holed as follows: (1) l = 1 X and h = 1 X , thus h = l = 1 X and hence(X, * , f ) l = (X, †, h) h . (2) l = 1 X and h = l, thus h = l = 1 X and hence (X, * , f ) l = (X, †, h) h . (3) l = h and h = 1 X , thus l = h = 1 X and hence (X, * , f ) l = (X, †, h) h . (4) l = h and h = l, thus (X, * , f ) l = (X, †, h) h .
Then (X, * , f ) has intuitionistic Γ-Fuzzy d-algebra say A = (m A , h A ) with Table 15.
Therefore (X, †, c) is edge, but (X, * , f ) is not edge. That means they have different structures. Further, the normalized Euclidean distance between their intuitionistic Γ-Fuzzy d-algebras A = (m A , h A ) and B = (m B , h B ) is = 0.03125.

Conclusions
In the present paper, we have introduced the concept of Γ-Fuzzy d-algebra and investigated some of its applications and essential properties. We think this work would enhance the scope for further study in this field of Fuzzy d-algebra. It is our hope that this work is going to impact the upcoming research works in this field of algebras with a new horizon of interest and innovation such as Γ-Fuzzy d-subalgebra, Γ-Fuzzy d-ideal, Γ-Fuzzy d # -ideal and Γ-Fuzzy d*-ideal. These new notions will depend on the structure of d-algebra and hence we can study their properties using the structure of d-algebra.

Disclosure statement
No potential conflict of interest was reported by the author.