Generalization of Fuzzy Soft BCK/BCI-Algebras

In this paper, the notions of (∈, ∈∨q)-fuzzy soft BCK/BCI-algebras and (∈, ∈∨q)-fuzzy soft sub-BCK/BCI-algebras are introduced, and related properties are investigated. Furthermore, relations between fuzzy soft BCK/BCI-algebras and (∈, ∈∨q)-fuzzy soft BCK/BCI-algebras are displayed. Moreover, conditions for an (∈, ∈∨q)-fuzzy soft BCK/BCI-algebra to be a fuzzy soft BCK/BCI-algebra are provided. Also, the union, the extended intersection, and the “AND”-operation of two (∈, ∈∨q)-fuzzy soft (sub-)BCK/BCI-algebras are discussed, and a characterization of an (∈, ∈∨q)-fuzzy soft BCK/BCI-algebra is established.


Introduction
e uncertainty which appeared in economics, engineering, environmental science, medical science, social science, and so on is too complicated to be captured within a traditional mathematical framework. In order to overcome this situation, a number of approaches including fuzzy set theory [1,2], probability theory, rough set theory [3,4], vague set theory [5], and the interval mathematics [6] have been developed. e concept of soft set was introduced by Molodtsov [7] as a new mathematical method to deal with uncertainties free from the errors being occurred in the existing theories. Later, Maji et al. [8,9] defined fuzzy soft sets and also described how soft set theory is applied to the problem of decision making. Study on the soft set theory is currently moving forward quickly. In [10], Jun et al. discussed the intersection-soft filters in R 0 -algberas. Roh and Jun [11] studied positive implicative ideals of BCK-algebras based on intersectional soft sets. Roy and Mayi [12] gave results on applying fuzzy soft sets to the problem of decision making. Aygünoǧlu and Aygün [13] proposed and investigated the notion of a fuzzy soft group. Furthermore, Jun et al. [14] applied the theory of fuzzy soft sets to BCK/BCI-algebras and introduced the notion of fuzzy soft BCK/BCI-algebras (briefly, FSB-algebras) and related notions. Moreover, Muhiuddin et al. studied and applied the soft set theory to the different algebraic structures on various aspects (see, e.g., [15][16][17][18][19][20][21][22][23]). Also, some related concepts based on the present work are studied in [24][25][26][27][28][29][30][31][32][33].
In this paper, we define the notions of (∈, ∈∨q)-FSBalgebras and (∈, ∈ ∨ q)-fuzzy soft sub-BCK/BCI-algebras. Further, we investigate related properties and consider relations between fuzzy soft BCK/BCI-algebras and (∈, ∈∨q)-fuzzy soft BCK/BCI-algebras. Moreover, we prove that every FSB-algebra over X is an (∈, ∈∨q)-FSB-algebra over X and also show by an example that the converse of the aforesaid statement is not true in general. In fact, we provide a condition for an (∈, ∈∨q)-FSB-algebra to be a FSB-algebra. In addition, we discuss the union, the extended intersection, and the "AND"-operation of two (∈, ∈∨q)-FSB-algebras. Finally, we establish a characterization of an (∈, ∈∨q)-fuzzy soft BCK/BCI-algebra. e paper is organized as follows. Section 2 summarizes some definitions and properties related to BCK/BCI-algebras, fuzzy sets, soft sets, and fuzzy soft sets which are needed to develop our main results. In Section 3, the notions of FSB-algebras are studied and the concepts of θ-identity and θ-absolute FSB-algebras are introduced. Section 4 is devoted to the study of (∈, ∈∨q)-FSBalgebra. e paper ends with a conclusion and a list of references.

Preliminaries
A BCK/BCI-algebra is the most important class of logical algebras which was introduced by K. Iséki.
In a BCK/BCI-algebra X, a nonempty subset T of X is called a BCK/BCI-subalgebra of X if ϖ * ϱ ∈ T ∀ ϖ, ϱ ∈ T.
For an initial universe set U and a set of parameters E, let P(U) denote the power set of U and Ω ⊂ E. Molodtsov [7] defined the soft set as follows.
Definition 1 (see [7]). A pair (ζ, Ω) is called a soft set over U, where ζ is a function given by ζ: Ω ⟶ P(U). (3) e set ζ(ε) for ε ∈ Ω may be considered as the set of ε-approximate elements of the soft set (ζ, Ω). Clearly, a soft set is not a set. We refer the reader to [7] for illustration where several examples are presented.
Let F(U) denote the set of all fuzzy sets in U.
Definition 2 (see [9]). A pair (ζ, Ω) is called a fuzzy soft set over U where ζ is a mapping given by For all ϖ ∈ Ω, ζ[ϖ] ∈ F(U) and it is called fuzzy value set of parameter ϖ. If ζ[ϖ], for all ϖ ∈ Ω, is a crisp subset of U, then (ζ, Ω) is degenerated to be the standard soft set.
us, fuzzy soft sets are a generalization of standard soft sets. We will use FS(U) to denote the set of all fuzzy soft sets over U.
e notion FS BCK/BCI A(X) will be used for the set of all (∈, ∈∨q)-fuzzy soft BCK/BCI-algebras.
where υ is any parameter in Ω.
e proof is followed from eorem 6 and Definition 9. Proof.
e proof is followed from eorem 7 and Definition 9.
We hope that this work will provide a deep impact on the upcoming research in this field and other soft algebraic studies to open up new horizons of interest and innovations. To extend these results, one can further study these notions on different algebras such as rings, hemirings, LA-semigroups, semihypergroups, semihyperrings, BL-algebras, MTL-algebras, R 0 -algebras, MValgebras, EQ-algebras, d-algebras, Q-algebras, and lattice implication algebras. Some important issues for future work are (1) to develop strategies for obtaining more valuable results and (2) to apply these notions and results for studying related notions in other algebraic (soft) structures.

Data Availability
No data were used to support the study.

Conflicts of Interest
e authors declare that they have no conflicts of interest. 6 Journal of Mathematics