Bipolar fuzzy soft subalgebras and ideals of BCK / BCI -algebras based on bipolar fuzzy points

Abstract

In this paper, the concept of quasi-coincidence of a bipolar fuzzy point within a bipolar fuzzy set is introduced. The notion of ∈-bipolar fuzzy soft set and q-bipolar fuzzy soft set is introduced based on a bipolar fuzzy set and characterizations for an ∈-bipolar fuzzy soft set and a q-bipolar fuzzy soft set to be bipolar fuzzy soft BCK/BCI-algebras are given. Also, the notion of (∈ , ∈ ∨ q)-bipolar fuzzy subalgebras and ideals are introduced and characterizes for an ∈-bipolar fuzzy soft set and q-bipolar fuzzy soft set to be a bipolar fuzzy soft BCK/BCI-algebras are established. Some characterization theorems of these (∈ , ∈ ∨ q)-bipolar fuzzy soft subalgebras and ideals are derived. The relationship among these (∈ , ∈ ∨ q)-bipolar fuzzy soft subalgebras and ideals are also considered.

Keywords

soft set ∈-bipolar fuzzy soft set

1 Introduction

Classical methods failed to solve the complicated problems successfully in engineering, economics, and environment, because of various typical for those problems. There are three theories: theory of probability, theory of fuzzy sets and the interval mathematics which are considered as mathematical tools for dealing with uncertainties. But all of these theories have their own difficulties that are pointed out by Molodtsov [1, 2]. Maji et al. [3] and Molodtsov [1] suggested that one reason for these difficulties may be due to inadequacy of the parametrization of the theory. To solve these difficulties, Molodtsov [1] introduced the concept of a soft set theory as a new mathematical tool for dealing with uncertainties which is free from difficulties and also, pointed out for the applications of soft sets in several directions. Maji et al. [4] described the application of soft set theory in a decision making problem. They also studied several operations in soft set theory [3].

Recently in many domains, we are able to deal with bipolar information. It is noted that positive information indicates what is granted to be possible, while negative information indicates what is considered to be impossible. This ideas has been recently motivated new research in several directions. In particular, fuzzy and possibilistic formalisms for bipolar information have been recently proposed [5]. When we deal with spatial information in image processing or in spatial reasoning, in this case bipolarity also occurs. When projected a satellite in space, then we may have set of positive information about the possible paths and may have set of negative information about the impossible paths of the satellite. If we considered rainy weather of a certain place, then there are some positive situation to may occur rain in that place, also may have some negative situation impossible to occur rain in that place. These corresponds to the idea that the union of positive and negative information does not cover the whole space. In 1994, Zhang [6 –8] first initiated the concept of bipolar fuzzy sets as a generalization of fuzzy sets [9]. Also, bipolar-valued fuzzy sets, which are introduced by Lee [10, 11], are an extension of fuzzy sets whose membership degree range is enlarged from the interval [0, 1] to [-1, 1]. In a bipolar fuzzy set, the membership degree 0 of an element means that the element is irrelevant to the corresponding property, the membership degree (0, 1] of an element indicates that the element somewhat satisfies the property, and the membership degree [-1, 0) of an element indicates that the element somewhat satisfies the implicit counter-property. Although bipolar fuzzy sets and intuitionistic fuzzy sets are similar, but they are differentsets [11].

It is well known that BCK/BCI-algebras are two classes of algebras of logic. The study of these algebras were initiated by Imai and Iseki [12, 13] in 1966 as a generalization of the concept of set-theoretic difference and propositional calculi. The study of structures of fuzzy sets in BCK/BCI-algebraic structure carried out by many researchers [14 –24]. Jana et al. [25 –35], and Jana and Pal [21] and Bej and Pal [36] has been done lot of works on BCK/BCI-algebras and B/BG/G-algebras which is related to these algebras.

Murali [37] introduced the definition of a fuzzy point belonging to fuzzy subset under a natural equivalence on a fuzzy subset. The idea of fuzzy set theory applied to a quasi-coincidence of a fuzzy point, is mentioned in [38], played a vital role to derived some types of fuzzy subsets. Bhakat and Das [39, 40] initiated the concept (α, β)-fuzzy subgroups by using the ‘belongs to’ relation (∈) and ‘quasi-coincident with’ relation (q) between a fuzzy point and a fuzzy subgroup, and introduced the concept of an (∈ , ∈ ∨ q)-fuzzy subgroup. Since (∈ , ∈ ∨ q)-fuzzy subgroup is an important generalization of Rosenfeld’s [41] fuzzy subgroup. Similar type of generalizations of the fuzzy subsystem can be made to the other algebraic structures [42 –47]. Jun et al. [48, 49] introduced the concept of (α, β)-fuzzy subalgebras and ideals and investigated their related properties. Ma et al. [50] introduced some kinds of (∈ , ∈ ∨ q)-interval-valued fuzzy ideals ofBCI-algebras.

Using the notion of bipolar-valued fuzzy sets,Lee [51] discussed bipolar fuzzy subalgebras and ideals of BCK/BCI-algebras. Also, Lee and Jun [17] discuss bipolar fuzzy a-ideals in BCK/BCI-algebras. Jun [52 –54] applied the notion of soft sets to the theory of BCK/BCI-algebras and d-algebras, and introduced the notion of soft BCK/BCI-algebras, soft subalgebras and soft d-algebras, and then described their basic properties. Jun et al. [55] introduced the notion of soft p-ideals and p-ideas of soft BCI-algebras and developed their basic properties. The algebraic structure of set theories dealing with uncertainties has been introduced by some authors. Aktas and Cagman [56] studied the basic concepts of soft set theory, and compared soft sets to fuzzy and rough sets, providing examples to clarify their differences and they defined the notion of fuzzy soft groups. Applications of soft set theory in real life problems are now given new momentum due to the general nature parametrization expressed by soft set. Recently, To the best of our knowledge no works are available on bipolar fuzzy soft sets in BCK/BCI-algebras. For this reason we are motivated to developed the theories on bipolar fuzzy soft sets in BCK/BCI-algebras.

In this paper, we introduced the notion of an ∈-bipolar fuzzy soft set and a q-bipolar fuzzy soft set based on a bipolar fuzzy set and provide characterizations for an ∈-bipolar fuzzy soft set and a q-bipolar fuzzy soft set to be bipolar fuzzy soft BCK/BCI-algebras. Using the notion of (∈ , ∈ ∨ q)-bipolar fuzzy BCK/BCI-subalgebras/ideals, we established characterizations for an ∈-bipolar fuzzy soft set and a q-bipolar fuzzy soft set to be bipolar fuzzy soft BCK/BCI-algebras.

The remainder of this article is structured as follows: Section 2 proceeds with a recapitulation of all required definitions and properties. In Section 3, concepts and operations of (∈ , ∈ ∨ q)-bipolar fuzzy soft subalgebras of BCK/BCI-algebras are proposed and discussed their properties in details. In Section 4, properties of (∈ , ∈ ∨ q)-bipolar fuzzy soft ideals re investigated. Finally, in Section 5, conclusion and scope for future research are given.

2 Preliminaries

In this section, some elementary aspects that are necessary for this paper are included.

By a BCI-algebra we mean an algebra (X, ∗ , 0) of type (2, 0) satisfying the following axioms for all x, y, z ∈ X:

(i) ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) =0

(ii) (x ∗ (x ∗ y)) ∗ y = 0

(iii) x ∗ x = 0

(iv) x ∗ y = 0 and y ∗ x = 0 imply x = y.

We can define a partial ordering “≤ " by x ≤ y if and only if x ∗ y = 0.

If a BCI-algebra X satisfies 0 ∗ x = 0 for all x ∈ X, then we say that X is a BCK-algebra. Any BCK-algebra X satisfies the following axioms for all x, y, z ∈ X:

(1) (x ∗ y) ∗ z = (x ∗ z) ∗ y

(2) ((x ∗ z) ∗ (y ∗ z)) ∗ (x ∗ y) =0

(3) x ∗ 0 = x

(4) x ∗ y = 0 ⇒ (x ∗ z) ∗ (y ∗ z) =0, (z ∗ y) ∗ (z ∗ x) =0 .

Throughout this paper, X always means a BCK/BCI-algebra without any specification.

A non-empty subset S of X is called a subalgebra of X if x ∗ y ∈ S for any x, y ∈ S. A nonempty subset I of X is called an ideal of X if it satisfies

(I₁) 0 ∈ I and

(I₂) x * y ∈ I and y ∈ I imply x ∈ I.

We refer the reader to the books [18, 57] for further information regarding BCK/BCI-algebras. A fuzzy set μ in a set is of the form $μ (y) = {\begin{matrix} t \in (0, 1] & if y = x \\ 0 & if y \neq x . \end{matrix}$ is said to be a fuzzy point with support x and value t and is denoted by x_t . For a fuzzy point x_t and a fuzzy set μ of a set X, Pu and Liu [38] gave meaning to the symbol x_tΦμ, where Φ ∈ {∈ , q, ∈ ∨ q, ∧ q}. To say that x_t ∈ μ (respectively, x_tqμ) means that μ (x) ≥ t (respectively μ (x) + t > 1), and in this case, x_t is said to belong to (respectively, be quasi-coincident with) a fuzzy set μ. To say that x_t ∈ ∨ qμ (respectively, x_t ∈ ∧ qμ) means that x_t ∈ μ or x_tqμ (respectively, x_t ∈ μ and x_tqμ). To say that $x_{t} \bar{Φ} μ$ means that x_tΦμ does not hold, where Φ ∈ {∈ , q, ∈ ∨ q, ∈ ∧ q}.

A fuzzy set μ in a BCK/BCI-algebra X is said to be a fuzzy subalgebra of X if it satisfies μ (x ∗ y) ≥ min {μ (x) , μ (y)} for all x, y ∈ X.

A fuzzy set μ of X is said to be a fuzzy ideal of X if it satisfies (i) μ (0) ≥ μ (x)) and (ii) μ (x) ≥ min {μ (x ∗ y) , μ (y)}, for all x, y ∈ X.

Proposition 2.1. [48] A fuzzy set μ of X is called a fuzzy subalgebra of X if and only if it satisfies x_t ∈ μ, y_s ∈ μ ⇒ (x ∗ y) _min(t,s) ∈ μ for all x, y ∈ X and t, s ∈ (0, 1] .

Proposition 2.2 [49] A fuzzy set μ of X is called a fuzzy ideal of X if and only if it satisfies (i) x_t ∈ μ ⇒ 0_t ∈ μ, (ii) (x ∗ y) _t ∈ μ, y_s ∈ μ ⇒ x_min(t,s) ∈ μ, for all x, y ∈ X and t, s ∈ (0, 1].

Definition 2.3. [10] A bipolar fuzzy set μ of X is defined as μ = {(x, μ^P (x) , μ^N (x)) : x ∈ X} where μ^P : X → [0, 1] and μ^N : X → [-1, 0] are mappings. The positive membership degree μ^P (x) denote the satisfaction degree of an element x to the property corresponding to a bipolar fuzzy set μ = {(x, μ^P (x) , μ^N (x) : x ∈ X} and the negative membership degree μ^N (x) denotes the satisfaction degree of an element x to some implicit counter property of μ = {(x, μ^P (x) , μ^N (x) : x ∈ X}. If μ^P (x) ≠0 and μ^N (x) =0 in this case it is regarded as having only positive satisfaction degree for μ = {(x, μ^P (x) , μ^N (x) : x ∈ X}. If μ^P (x) =0 and μ^N (x) ≠0, in this case it is regarded that x does not satisfy the property of μ = {(x, μ^P (x) , μ^N (x) : x ∈ X}, but somewhat satisfies the counter-property of μ = {(x, μ^P (x) , μ^N (x) : x ∈ X}. Some case it is possible for an element x to be μ^P (x) ≠0 and μ^N (x) ≠0 when the membership function of the property overlaps that of its counter-property of its some portion of domain [11]. Simply, we shall use the symbol μ = (μ^P, μ^N) for the bipolar fuzzy set μ = {(x, μ^P (x) , μ^N (x)) |x ∈ X}.

Definition 2.4. [7] For every two bipolar fuzzy sets $A = (μ_{A}^{P}, μ_{A}^{N})$ and $B = (μ_{B}^{P}, μ_{B}^{N})$ in X, we define $(A \cup B) (x) = {max {μ_{A}^{P} (x), μ_{B}^{P} (x)}, min {μ_{A}^{N} (x), μ_{B}^{N} (x)}}$ $(A \cap B) (x) = {min {μ_{A}^{P} (x), μ_{B}^{P} (x)}, max {μ_{A}^{N} (x), μ_{B}^{N} (x)}} .$

Proposition 2.5. [51] A bipolar fuzzy set μ = (μ^P, μ^N) of X is called a bipolar fuzzy subalgebras of X if it satisfies μ^P (x ∗ y) ≥ min {μ^P (x) , μ^P (y)} and μ^N (x ∗ y) ≤ max {μ^N (x) , μ^N (y)} for all x, y ∈ X.

Definition 2.6. [51] A bipolar fuzzy set μ = (μ^P, μ^N) of X is called a bipolar fuzzy ideal of X if it satisfies the following assertions

(i) μ^P (0) ≥ μ^P (x) and μ^N (0) ≤ μ^N (x)

(ii) μ^P (x) ≥ min {μ^P (x ∗ y) , μ^P (y)}

(iii) μ^N (x) ≤ max {μ^N (x ∗ y) , μ^N (y)} for all x, y ∈ X.

Definition 2.7. [34] A bipolar fuzzy set $A = (μ_{A}^{P}, μ_{A}^{N})$ of X is called a bipolar fuzzy subalgebras of X if and only if the following assertion is valid $x_{t} \in μ_{A}^{P}, y_{s} \in μ_{A}^{P} \Rightarrow (x * y)_{min (t, s)} \in μ_{A}^{P}$ and $x_{m} \in μ_{A}^{N}, y_{n} \in μ_{A}^{N} \Rightarrow (x * y)_{max (m, n)} \in μ_{A}^{N}$ , for all x, y ∈ X, t, s ∈ (0, 1] and m, n ∈ [-1, 0).

Theorem 2.8. [34] A bipolar fuzzy set $A = (μ_{A}^{P}, μ_{A}^{N})$ of X is an (∈ , ∈ ∨ q)-bipolar fuzzy subalgebra of X if and only if it satisfies $μ_{A}^{P} (x * y) \geq min {μ_{A}^{P} (x), μ_{A}^{P} (y), 0.5}$ and $μ_{A}^{N} (x * y) \leq max {μ_{A}^{N} (x), μ_{A}^{N} (y), - 0.5}$ for all x, y ∈ X.

Theorem 2.9. [34] A bipolar fuzzy set $A = (μ_{A}^{P}, μ_{A}^{N})$ of X is called a bipolar fuzzy ideal of X if and only if the following assertions is valid

(i) $x_{t} \in μ_{A}^{P} \Rightarrow 0_{t} \in μ_{A}^{P}$ and $x_{m} \in μ_{A}^{N} \Rightarrow 0_{m} \in μ_{A}^{N}$ , for all x ∈ X, t ∈ [0, 1], m ∈ [-1, 0]) (ii) $(x * y)_{t} \in μ_{A}^{P}, y_{s} \in μ_{A}^{P} \Rightarrow x_{min (t, s)} \in μ_{A}^{P}$ , for all x, y ∈ X, t, s ∈ [0, 1] (iii) $(x * y)_{m} \in μ_{A}^{N}, y_{n} \in μ_{A}^{N} \Rightarrow x_{max (m, n)} \in μ_{A}^{N}$ , for all x, y ∈ X, m, n ∈ [-1, 0]).

Proposition 2.10. [34] A bipolar fuzzy set $A = (μ_{A}^{P}, μ_{A}^{N})$ of X is an (∈ , ∈ ∨ q)-bipolar fuzzy ideal of X if and only if it satisfies the following conditions: (i) $μ_{A}^{P} (0) \geq min {μ_{A}^{P} (x), 0.5}$ and $μ_{A}^{N} (0) \leq max {μ_{A}^{N} (x), - 0.5}$ for all x ∈ X (ii) $μ_{A}^{P} (x) \geq min {μ_{A}^{P} (x * y), μ_{A}^{P} (y), 0.5}$ for all x, y ∈ X (iii) $μ_{A}^{N} (x) \leq max {μ_{A}^{N} (x * y), μ_{A}^{N} (y), - 0.5}$ for all x, y ∈ X.

Molodtsov [1] defined the soft set in the following way: Let U be an initial universe set and E be a set of parameters. Let P (U) denote power set of U and A ⊂ E.

Definition 2.11. [1] A pair (ϑ, A) is called a soft set over U and ϑ is a mapping given by $ϑ : A \to P (U) .$

In other words, a soft set over U is a parameterized family of subsets of the universe U. For any element ∈ of A, ϑ (∈) may be considered as the set of ∈-approximate element of the soft set (ϑ, A). Given a fuzzy set μ of X and A ⊆ [0, 1], we define the two set valued functions ϑ : A → P (X) and ϑ_q : A → P (X) by ϑ (t) = {x ∈ X|x_t ∈ μ} and ϑ_q (t) = {x ∈ X|x_tqμ} for all t ∈ A, respectively. Then (ϑ, A) and (ϑ_q, A) are two soft sets of X, which are called an ∈-soft set and a q-soft set over X respectively.

Definition 2.12. [51] Let U be an initial universe and A ⊆ E be a set of parameters. Let BF (U) denote set of all bipolar fuzzy soft sets of U. A pair $(F, A)$ is called a bipolar fuzzy soft set over U, where F is a mapping given by F : A → BF (U). Thus a bipolar fuzzy soft set over U is a parameterized family of bipolar fuzzy subsets of the universe U. For any ∈ ∈ A, $F (\in) = {x, μ_{A (\in)}^{P}, μ_{A (\neg \in)}^{N}}$ where $μ_{A (\in)}^{P} : U \to (0, 1]$ and $μ_{A (\neg \in)}^{N} : U \to [- 1, 0)$ are mappings. For any ∈ in A, F (∈) is referred to as the set of ∈-approximate elements of the bipolar fuzzy soft set $(F, A)$ , where $μ_{A (\in)}^{P}$ denotes the degree of x keeping the parameter ∈, $μ_{A (\in)}^{N}$ denotes the degree of x keeping the non-parameter ∈. □

3 Bipolar fuzzy points and bipolar fuzzy soft BCK/BCI-algebras

Let us now define bipolar fuzzy points of X, $(F [μ], A)$ as a bipolar fuzzy ∈-soft set, $(F_{q} [μ], A)$ as bipolar fuzzy q-soft set unless otherwise specified.

Definition 3.1. Let μ = (μ^P, μ^N) be a bipolar fuzzy set in a set X is of the form $μ^{P} (y) = {\begin{matrix} t \in (0, 1] & if y = x \\ 0 & if y \neq x . \end{matrix}$

$μ^{N} (y) = {\begin{matrix} m \in [- 1, 0) & if y = x \\ 0 & if y \neq x . \end{matrix}$ is said to be together (x, x_t, x_m) as bipolar fuzzy point with support x and values t and m of x_t and x_m respectively. For a bipolar fuzzy point (x, x_t, x_m) and a bipolar fuzzy set μ = (μ^P, μ^N) in a set X, we give the meaning of the symbol (x_tΦμ^P, x_mΦμ^N), where Φ ∈ {∈ , q, ∈ ∨ q, ∈ ∧ q} . To say that x_t ∈ μ^P (respectively, x_tqμ^P) and x_m ∈ μ^N (respectively, x_mqμ^N) means that μ^P (x) ≥ t (respectively, μ^P (x) + t > 1) and μ^N (x) ≤ m (respectively, μ^N (x) + m < -1), and in this case we say that, x_t is said to belong to (respectively, be quasi-coincident with) and x_m is said to belong to (respectively, be quasi-coincident with) a bipolar fuzzy set μ = (x, μ^P, μ^N). To say that x_t ∈ ∨ q (respectively, x_t ∈ ∧ q) and x_m ∈ ∨ q (respectively, x_m ∈ ∧ q) means that x_t ∈ μ^P or x_tqμ^P (respectively, x_t ∈ μ^P and x_tqμ^P) and x_m ∈ μ^N or x_mqμ^N (respectively, x_m ∈ μ^N and x_mqμ^N). To say that $(x_{t} \bar{Φ} μ^{P}, x_{m} \bar{Φ} μ^{N})$ means that x_tΦμ^P does not hold and x_mΦμ^N does not hold, where Φ ∈ {∈ , q, ∈ ∨ q, ∈ ∧ q}.

Definition 3.2. Let μ = (μ^P, μ^N) be a bipolar fuzzy set of X and (m, t) ∈ [-1, 0] × [0, 1]. We define U (μ^P ; t) = {x ∈ X|μ^P (x) ≥ t} is called a t-level cut set of μ^P and U (μ^N ; m) = {x ∈ X|μ^N (x) ≤ m} is called m-level cut set of μ^N of the bipolar fuzzy set μ = (μ^P, μ^N).

Definition 3.3. Let $(F [μ], A)$ be a bipolar fuzzy soft set over X. Then $(F [μ], A)$ is called a bipolar fuzzy soft subalgebra over X if $F [μ] (x)$ is a bipolar fuzzy subalgebras of X for all x ∈ A, that is, a bipolar fuzzy soft set $(F [μ], A)$ is called a bipolar fuzzy soft subalgebras of X if satisfies two conditions μ^P (x ∗ y) ≥ min {μ^P (x) , μ^P (y)} and μ^N (x ∗ y) ≤ max {μ^N (x) , μ^N (y)} .

For our convenience the empty set ∅ is regarded as a bipolar fuzzy subalgebras of X.

Example 3.4. Consider a BCI-algebra X = {0, a, 1, 2, 3} with the following Cayley table: $\begin{matrix} * & 0 a 123 \\ 0 & 00321 \\ a & a 0321 \\ 1 & 11032 \\ 2 & 22103 \\ 3 & 33210 \end{matrix}$ and $(F [μ], A)$ be a bipolar fuzzy soft set over X, where A = (0, 1] and ¬A = [-1, 0) and $F : A \to P (X)$ and $F : \neg A \to P (X)$ are a set-valued functions defined by $F [μ^{P}] (t) = {\begin{matrix} \emptyset, & if 0.8 < t \\ {0}, & if 0.6 < t \leq 0.8 \\ {0, a}, & if 0.3 < t \leq 0.6 \\ X, & if 0 \leq t \leq 0.3 . \end{matrix}$ and $F [μ^{N}] (s) = {\begin{matrix} \emptyset, & if - 1 \leq s < - 0.8 \\ {0}, & if - 0.7 \leq s < - 0.5 \\ {0, a}, & if - 0.5 \leq s < - 0.2 \\ X, & if - 0.2 \leq s < 0 . \end{matrix}$ Then, $F [μ] (x)$ is a bipolar fuzzy BCI-subalgebra of X for all x ∈ A.

Example 3.5. Let X = {0, a, b, c, d} be a BCK-algebra with the following Cayley table: $\begin{matrix} * & 0 a b c d \\ 0 & 00000 \\ a & a 0 a 00 \\ b & b b 000 \\ c & c c c 00 \\ d & d c d a 0 \end{matrix}$ Let us define the set-valued functions such that $F [μ^{P}] (t) = {\begin{matrix} \emptyset, & if 0.7 < t \\ {0, b}, & if 0.2 < t \leq 0.7 \\ X, & if 0 \leq t \leq 0.2 . \end{matrix}$ and $F [μ^{N}] (s) = {\begin{matrix} \emptyset, & if - 1 < s < - 0.8 \\ {0, b}, & if - 0.8 \leq s < - 0.7 \\ X, & if - 0.7 \leq s < 0 . \end{matrix}$

In this example, $F [μ] (x)$ is a bipolar fuzzy subalgebra of X for all x ∈ A, and so $(F [μ], A)$ is a bipolar fuzzy soft subalgebra over X.

Given a bipolar fuzzy set μ = (μ^P, μ^N) of X, A ⊆ (0, 1] and ¬A ⊆ [-1, 0). Consider four set-valued functions $\begin{matrix} F [μ^{P}] : A \to P (X), t \mapsto {x \in X | x_{t} \in μ^{P}}, \\ F_{q} [μ^{P}] : A \to P (X), t \mapsto {x \in X | x_{t} q μ^{P}}, \\ F [μ^{N}] : \neg A \to P (X), m \mapsto {x \in X | x_{m} \in μ^{N}} and \\ F_{q} [μ^{N}] : \neg A \to P (X), m \mapsto {x \in X | x_{m} q μ^{N}} . \end{matrix}$ Then $(F [μ^{P}], F [μ^{N}], A)$ and $(F_{q} [μ^{P}], F_{q} [μ^{N}], A)$ , or precisely, $(F [μ], A)$ and $(F_{q} [μ], A)$ are bipolar fuzzy ∈-soft set and q-soft set respectively.

Theorem 3.6. Let μ = (μ^P, μ^N) be a bipolar fuzzy set over X. Then $(F [μ], A)$ is called bipolar fuzzy ∈-soft set over X with A = (0, 1] and ¬A = [-1, 0). Then $(F [μ], A)$ is a bipolar fuzzy soft subalgebra over X if and only if μ is a bipolar fuzzy subalgebras of X.

Proof: Assume that $(F [μ], A)$ be a bipolar fuzzy soft subalgebra over X. If μ is not a bipolar fuzzy subalgebras of X, then there exist a, b ∈ X such that μ^P (a ∗ b) < min {μ^P (a) , μ^P (b)} and μ^N (a ∗ b) > max {μ^N (a) , μ^N (b)}. Take t ∈ A and m ∈ ¬ A such that μ^P (a ∗ b) < t ≤ min {μ^P (a) , μ^P (b)} and μ^N (a ∗ b) > m ≥ max {μ^N (a) , μ^N (b)}. Then a_t ∈ μ^P and b_t ∈ μ^P, but (a ∗ b) _min(t,t) = (a ∗ b) _t ∉ μ^P, and a_m ∈ μ^N, b_m ∈ μ^N but (a ∗ b) _max(m,m) = (a ∗ b) _m ∉ μ^N. Hence, $a, b \in F [μ^{P}] (t)$ and $a, b \in F [μ^{N}] (m)$ . But $a * b \notin F [μ^{P}] (t)$ and $a * b \notin F [μ^{N}] (m)$ . This is a contradiction. Therefore, μ^P (x ∗ y) ≥ min {μ^P (x) , μ^P (y)} and μ^N (x ∗ y) ≤ max {μ^N (x) , μ^N (y)} for all x, y ∈ X.

Conversely, suppose that μ is a bipolar fuzzy subalgebras of X. Let t ∈ A, m ∈ ¬ A, and $x, y \in F [μ^{P}] (t),$ and $x, y \in F [μ^{N}] (m)$ . Then x_t ∈ μ^P, y_t ∈ μ^P and also, x_m ∈ μ^N and y_m ∈ μ^N. It follows from Definition 2.7 that (x ∗ y) _t = (x ∗ y) _min(t,t) ∈ μ^P and (x ∗ y) _m = (x ∗ y) _max(m,m) ∈ μ^N so that $x * y \in F [μ^{P}] (t)$ and $x * y \in F [μ^{N}] (m)$ . Thus, F [μ] is a bipolar fuzzy subalgebras of X, i.e., $(F, A)$ is a bipolar fuzzy soft subalgebra over X. □

Theorem 3.7. Let μ = (μ^P, μ^N) be a bipolar fuzzy set of X. Then $(F_{q} [μ], A)$ is bipolar fuzzy q-soft set over X with A = (0, 1] and ¬A = [-1, 0). Then $(F_{q} [μ], A)$ is a bipolar fuzzy q-soft BCK/BCI-algebra over X if and only if μ is a bipolar fuzzy subalgebras of X.

Proof: Suppose that μ is a bipolar fuzzy subalgebras of X. Let t ∈ A, m ∈ ¬ A and x, y ∈ F_q [μ] (t, m). Then x_tqμ^P, y_tqμ^P and x_mqμ^N, y_mqμ^N hold, i.e.,μ^P (x) + t > 1, μ^P (y) + t > 1 and μ^N (x) + m < -1, μ^N (y) + m < -1. It follows from Proposition 2.5 that μ^P (x ∗ y) + t ≥ min {μ^P (x) , μ^P (y)} + t = min {μ^P (x) + t, μ^P (y) + t} >1 and μ^N (x ∗ y) + m ≤ max {μ^N (x) , μ^N (y)} + m = max {μ^N (x) + m, μ^N (y) + m} < -1, so that (x ∗ y) _tqμ^P and (x ∗ y) _mqμ^N, i.e., $x * y \in F_{q} [μ^{P}] (t)$ and $x * y \in F_{q} [μ^{N}] (m)$ . Hence, $(F_{q} [μ], A)$ is a bipolar fuzzy subalgebras of X for all t ∈ A and for all m ∈ ¬ A. Hence, $(F [μ], A)$ is a bipolar fuzzy soft subalgebra over X.

Conversely, assume that $(F_{q} [μ], A)$ is a bipolar fuzzy subalgebra over X. If μ^P (a ∗ b) < min {μ^P (a) , μ^P (b)} and μ^N (a ∗ b) > max {μ^N (a) , μ^N (b)} for some a, b ∈ X, then we can choose t ∈ A and m ∈ ¬ A such that μ^P (a ∗ b) + t ≤ 1 < min {μ^P (a) , μ^P (b)} + t and μ^N (a ∗ b) ≥ -1 > max {μ^N (a) , μ^N (b)} + m. Thena_tqμ^P, b_tqμ^P and a_mqμ^N, b_mqμ^N, but $(a * b)_{t} \bar{q} μ^{P}$ and $(a * b)_{m} \bar{q} μ^{N}$ i.e., $a_{t} \in F_{q} [μ^{P}] (t)$ , $b_{t} \in F_{q} [μ^{P}] (t)$ and $a_{m} \in F_{q} [μ^{N}] (m)$ , $b_{m} \in F_{q} [μ^{N}] (m)$ , but $a * b \notin F_{q} [μ^{P}] (t)$ and $a * b \notin F_{q} [μ^{N}] (m)$ , i.e., a ∗ b ∉ F_q [μ] (t, m). This is a contradiction. Therefore, μ is a bipolar fuzzy subalgebras of X. □

Theorem 3.8. Let μ = (μ^P, μ^N) be a bipolar fuzzy set of X and $(F [μ], A)$ be a bipolar fuzzy ∈-soft set over X with A = (0.5, 1] and ¬A = [-1, - 0.5) respectively. Then the following assertions are equivalent

(i) $(F [μ], A)$ is a bipolar fuzzy soft subalgebra over X

(ii) max {μ^P (x ∗ y) , 0.5} ≥ min {μ^P (x) , μ^P (y)} and min {μ^N (x ∗ y) , -0.5} ≤ max {μ^N (x) , μ^N (y)}, for all x, y ∈ X.

Proof: Assume that $(F [μ], A)$ is a bipolar fuzzy soft subalgebra over X. Then F [μ] is a bipolar fuzzy subalgebras of X for all t ∈ A and m ∈ ¬ A. If there exist a, b ∈ X such that max {μ^P (a ∗ b) , 0.5} < t = min {μ^P (a) , μ^P (b)} and min {μ^N (a ∗ b) , -0.5} > m = max {μ^N (a) , μ^N (b)}, then it follows that a_t ∈ μ^P, b_t ∈ μ^P and a_m ∈ μ^N, b_m ∈ μ^N but $(a * b)_{t} \bar{\in} μ^{P}$ and $(a * b)_{m} \bar{\in} μ^{N}$ . It follows that $a, b \in F [μ^{P}] (t)$ and $a, b \in F [μ^{N}] (m)$ but $a * b \notin F [μ^{P}] (t)$ and $a * b \notin F [μ^{N}] (m)$ . This is contradiction, and so max {μ^P (x ∗ y) , 0.5} ≥ min {μ^P (x) , μ^P (y)} and min {μ^N (x ∗ y) , -0.5} ≤ max {μ^N (x) , μ^N (y)}. Conversely, suppose that the condition (ii) is valid. Let t ∈ A, m ∈ ¬ A and $x, y \in F [μ^{P}] (t)$ , and $x, y \in F [μ^{N}] (m)$ . Then we have x_t ∈ μ^P, y_t ∈ μ^P and x_m ∈ μ^N, y_m ∈ μ^N, which equivalently, μ^P (x) ≥ t, μ^P (y) ≥ t and μ^N (x) ≤ m, μ^N (y) ≤ m. Hence, max {μ^P (x ∗ y) , 0.5} ≥ min {μ^P (x) , μ^P (y)} ≥ t > 0.5 and min {μ^N (x ∗ y) , -0.5} ≤ max {μ^N (x) , μ^N (y)} ≤ m < -0.5. and so, μ^P (x ∗ y) ≥ t and μ^N (x ∗ y) ≤ m i.e., (x ∗ y) _t ∈ μ^P and (x ∗ y) _m ∈ μ^N. Therefore, $x * y \in F [μ^{P}] (t)$ and $x * y \in F [μ^{N}] (m)$ which shows that $(F [μ], A)$ is a bipolar fuzzy soft subalgebra over X.□

Theorem 3.9. Let μ = (μ^P, μ^N) be a bipolar fuzzy set of X and $(F [μ], A)$ be a bipolar ∈-fuzzy soft set over X with A = (0, 0.5] and ¬A = [-0.5, 0). Then following conditions are equivalent

(i) μ is an (∈ , ∈ ∨ q)-bipolar fuzzy subalgebras of X

(ii) $(F [μ], A)$ is a bipolar fuzzy soft subalgebra over X.

Proof: Let us assume that μ is an (∈ , ∈ ∨ q)-bipolar fuzzy subalgebras of X. Let t ∈ A, m ∈ ¬ A and x, y ∈ F [μ] i.e., $x, y \in F [μ^{P}] (t)$ and $x, y \in F [μ^{N}] (m)$ . Then x_t ∈ μ^P, y_t ∈ μ^P and x_m ∈ μ^N, y_m ∈ μ^N, i.e., μ^P (x) ≥ t, μ^P (y) ≥ t and μ^N (x) ≤ m, μ^N (y) ≤ m. Then from Theorem 2.8 we get $\begin{matrix} μ^{P} (x * y) \geq \\ min {μ^{P} (x), μ^{P} (y), 0.5} \geq min {t, 0.5} = t \\ μ^{N} (x * y) \leq \\ max {μ^{N} (x), μ^{N} (y), - 0.5} \leq min {m, - 0.5} = m . \end{matrix}$

So that (x ∗ y) _t ∈ μ^P and (x ∗ y) _m ∈ μ^N or equivalently $x * y \in F [μ^{P}] (t)$ and $x * y \in F [μ^{N}] (m)$ i.e., x ∗ y ∈ F [μ]. Hence, $(F [μ], A)$ is a bipolar fuzzy soft subalgebra over X. Conversely, let us suppose that (ii) is valid. If there exist a, b ∈ X such that μ^P (a ∗ b) < min {μ^P (a) , μ^P (b) , 0.5} and μ^N (a ∗ b) > max {μ^N (a) , μ^N (b) , -0.5}, then we can choose that t ∈ (0, 1) and m ∈ (-1, 0) such that $μ^{P} (a * b) < t \leq min {μ^{P} (a), μ^{P} (b), 0.5}$ $μ^{N} (a * b) > m \geq max {μ^{N} (a), μ^{N} (b), - 0.5} .$ Thus t ≤ 0.5 and m ≥ -0.5, so that a_t ∈ μ^P, b_t ∈ μ^P and a_m ∈ μ^N, b_m ∈ μ^N i.e., $a \in F [μ^{P}] (t)$ , $b \in F [μ^{P}] (t)$ and $a \in F [μ^{N}] (m)$ , $b \in F [μ^{N}] (m)$ . Since F [μ] is a subalgebras of X, it follows that $a * b \in F [μ^{P}] (t)$ for all t ≤ 0.5 and $a * b \in F [μ^{N}] (m)$ for all m ≥ -0.5, so that (a ∗ b) _t ∈ μ^P or equivalently μ^P (a ∗ b) ≥ t for all t ≤ 0.5 and (a ∗ b) _m ∈ μ^N or equivalently μ^N (a ∗ b) ≤ m for all m ≥ -0.5. This is a contradiction. Hence, μ^P (x ∗ y) ≥ min {μ^P (x) , μ^P (y) , 0.5} and μ^N (x ∗ y) ≤ max {μ^N (x) , μ^N (y) , -0.5} for all x, y ∈ X. It follows from Theorem 2.8 that μ is an (∈ , ∈ ∨ q)-bipolar fuzzy subalgebras of X. □

Example 3.10. Consider a BCI-algebra X = {0, a, b, c} with the Caley following table

$\begin{matrix} * & 0 a b c \\ 0 & 0 a b c \\ a & a 0 c b \\ b & b c 0 a \\ c & c b a 0 \end{matrix}$ let us define the bipolar fuzzy set μ of X as follows that μ^P (0) =0.6, μ^P (a) =0.7, μ^P (b) = μ^P (c) =0.3 and μ^N (0) = -0.8, μ^N (a) = μ^N (c) = -0.3, μ^N (b) = -0.7. Then μ is an (∈ , ∈ ∨ q)-bipolar fuzzy BCI-subalgebras of X. We take A = (0, 0.5] and ¬A = [-0.5, 0), and let $(F [μ], A)$ be a bipolar ∈-fuzzy soft set over X. Then $F [μ^{P}] (t) = {\begin{matrix} X, & if t \in (0, 0.3] \\ {0, a}, & if t \in (0.3, 0.5] . \end{matrix}$ $F [μ^{N}] (s) = {\begin{matrix} X, & if s \in [- 0.3, 0) \\ {0, b}, & if s \in (- 0.3, - 0.5] . \end{matrix}$ which are bipolar BCI-fuzzy subalgebras of X. Hence, $(F [μ], A)$ is a bipolar fuzzy soft BCI-algebra over X.

4 Bipolar fuzzy soft BCK/BCI-ideals

Definition 4.1. Let $(F [μ], A)$ be a bipolar fuzzy soft set of X. Then $(F [μ], A)$ is called a bipolar fuzzy soft ideal over X if F [μ] (x) is a bipolar fuzzy ideal of X for all x ∈ A.

Example 4.2. Let X = {0, a, b, c, d} be a BCK-algebra with the following Cayley table: $\begin{matrix} * & 0 a b c d \\ 0 & 00000 \\ a & a 0 a 0 a \\ b & b b 0 b 0 \\ c & c a c 0 c \\ d & d d d d 0 \end{matrix}$ Let us define the set-valued functions such that $F [μ^{P}] (t) = {\begin{matrix} \emptyset, & if 0.7 < t \\ {0}, & if 0.3 < t \leq 0.7 \\ {0, a, c}, & if 0.2 < t \leq 0.3 \\ X, & if 0 \leq t \leq 0.2 . \end{matrix}$ and $F [μ^{N}] (s) = {\begin{matrix} \emptyset, & if - 1 \leq s < - 0.9 \\ {0}, & if - 0.9 \leq s < - 0.7 \\ {0, c}, & if - 0.7 \leq s < - 0.6 \\ {0, a, c}, & if - 0.6 \leq s < - 0.4 \\ {0, a, b, c}, & if - 0.4 \leq s < - 0.3 \\ X, & if - 0.3 \leq s < 0 . \end{matrix}$

In this example, $F [μ] (x)$ is a bipolar fuzzy ideal of X for all x ∈ A, and so $(F [μ], A)$ is a bipolar fuzzy soft ideal over X.

Theorem 4.3. Let μ = (μ^P, μ^N) be a bipolar fuzzy set of X and $(F [μ], A)$ be a bipolar fuzzy ∈-soft set over X with A = (0, 1] and ¬A = [-1, 0). Then $(F [μ], A)$ is a bipolar fuzzy soft ideal over X if and only if μ is a bipolar fuzzy ideal of X.

Proof: Suppose μ = (μ^P, μ^N) is a bipolar fuzzy BCK/BCI-algebra of X and let t, s ∈ A, m, n ∈ ¬ A. If x ∈ F [μ], then x_t ∈ μ^P and x_m ∈ μ^N. Then follows from Theorem 2.9 (i) that 0_t ∈ μ^P and 0_m ∈ μ^N, i.e., 0 ∈ F [μ] (t, m) i.e., $0 \in F [μ^{P}] (t)$ and $0 \in F [μ^{N}] (m)$ . Let x, y ∈ X be such that $(x * y) \in F [μ^{P}] (t)$ and $y \in F [μ^{P}] (t)$ and also, $(x * y) \in F [μ^{N}] (m)$ and $y \in F [μ^{N}] (m)$ . Then (x ∗ y) _t ∈ μ^P and y_t ∈ μ^P, and (x ∗ y) _m ∈ μ^N and y_m ∈ μ^N, which are imply from Theorem 2.9 (ii) and (iii) that x_t = x_min(t,t) ∈ μ^P and x_m = x_max(m,m) ∈ μ^N. Hence, we get $x \in F [μ^{P}] (t)$ and $x \in F [μ^{N}] (m)$ , and thus $(F [μ], A)$ is a bipolar fuzzy soft ideal over X.

Conversely, assume that $(F [μ], A)$ is a bipolar fuzzy soft ideal over X. If there exists a ∈ X such that μ^P (0) < μ^P (a) and μ^N (0) > μ^N (a), now we select t ∈ A and m ∈ ¬ A such that μ^P (0) < t ≤ μ^P (a) and μ^N (0) > m ≥ μ^N (a). Then 0_t ∉ μ^P and 0_m ∉ μ^N, i.e., $0 \notin F [μ^{P}] (t)$ and $0 \notin F [μ^{N}] (m)$ . This is a contradiction. Thus μ^P (0) ≥ μ^P (x) and μ^N (0) ≤ μ^N (x) for all x ∈ X. Suppose there exists a, b ∈ X such that μ^P (a) < min {μ^P (a ∗ b) , μ^P (b)} and μ^N (a) > max {μ^N (a ∗ b) , μ^N (b). Take s ∈ A and n ∈ ¬ A such that μ^P (a) < s ≤ min {μ^P (a ∗ b) , μ^N (b)} and μ^N (a) > n ≥ max {μ^N (a ∗ b) , μ^N (b)}. Then (a ∗ b) _s ∈ μ^P, b_s ∈ μ^P and (a ∗ b) _n ∈ μ^N, b_n ∈ μ^N, but a_s ∉ μ^P and a_n ∉ μ^N, i.e., $a * b \in F [μ^{P}] (s)$ and $b \in F [μ^{P}] (s)$ but $a_{s} \notin F [μ^{P}] (s)$ and $a * b \in F [μ^{N}] (n)$ , $b \in F [μ^{N}] (n)$ but $a \notin F [μ^{N}] (n)$ . This is a contradiction, so $μ^{P} (x) \geq min {μ^{P} (x * y), μ^{P} (y)}$ $μ^{N} (x) \leq max {μ^{N} (x * y), μ^{N} (y)}$ for all x, y ∈ X. Therefore, μ is a bipolar fuzzy ideal of X.□

Theorem 4.4. Let μ = (μ^P, μ^N) be a bipolar fuzzy set of X and $(F_{q} [μ], A)$ be a bipolar fuzzy q-soft set over X with A = (0, 1] and ¬A = [-1, 0). Then the following assertions are equivalent

(i) μ is a bipolar fuzzy ideal of X.

(ii) F_q [μ]≠ ∅ is a bipolar fuzzy ideal of X.

Proof: Assume that μ = (μ^P, μ^N) is a bipolar fuzzy BCK/BCI-ideal of X. Let t ∈ A and m ∈ ¬ A be such that $F_{q} [μ^{P}] (t) \neq \emptyset$ and $F_{q} [μ^{N}] (m) \neq \emptyset$ . If $0 \notin F_{q} [μ^{P}] (t)$ and $0 \notin F_{q} [μ^{N}] (m)$ , then $0_{t} \bar{q} μ^{P}$ and $x_{m} \bar{q} μ^{N}$ and so μ^P (0) + t < 1 and μ^N (0) + m > -1. It follows from Definition 2.6 (i) that μ^P (x) + t ≤ μ^P (0) + t < 1 and μ^N (x) + m ≥ μ^N (0) + m > -1 for all x ∈ X so we get $F_{q} [μ^{P}] (t) = 0$ and $F_{q} [μ^{N}] (m) = 0$ . This is a contradiction, therefore $0 \in F_{q} [μ^{P}] (t)$ and $0 \in F_{q} [μ^{N}] (m)$ , i.e, 0 ∈ F_q [μ] (t, m). Let x, y ∈ X be such that $x * y \in F_{q} [μ^{P}] (t)$ , $y \in F_{q} [μ^{P}] (t)$ and $x * y \in F_{q} [μ^{N}] (m)$ , $y \in F_{q} [μ^{N}] (m)$ . Then (x ∗ y) _tqμ^P, y_tqμ^P and (x ∗ y) _mqμ^N, y_mqμ^N, or equivalently, μ^P (x ∗ y) + t > 1 and μ^P (y) + t > 1, and μ^N (x ∗ y) + m < -1, μ^N (y) + m < -1. By Definition 2.6 (ii), (iii) we have

μ^P (x) + t ≥ min {μ^P (x ∗ y) , μ^P (y)} + t = min {μ^P (x ∗ y) + t, μ^P (y) + t} >1 and μ^N (x) + m ≤ max {μ^N (x ∗ y) , μ^N (y)} + m = max {μ^N (x ∗ y) + m, μ^N (y) + m} < -1,

and so x_tqμ^P and x_mqμ^N, i.e., $x \in F_{q} [μ^{P}] (t)$ and $x \in F_{q} [μ^{N}] (m)$ .Thus F_q [μ] (t, m) is a bipolar fuzzy BCK/BCI-ideal of X. Conversely, assume that (ii) is valid. If μ^P (0) < μ^P (a) and μ^N (0) > μ^N (a) for some a ∈ X, then μ^P (0) + t ≤ 1 < μ^P (a) + t and μ^N (0) + m ≥ -1 > μ^N (a) + m for some t ∈ A and m ∈ ¬ A. Thus a_tqμ^P and a_mqμ^N, and so $F_{q} [μ^{P}] (t) \neq \emptyset$ and $F_{q} [μ^{N}] (m) \neq \emptyset$ , i.e., F_q [μ] (t, m)≠ ∅. Hence $0 \in F_{q} [μ^{P}] (t)$ , and $0 \in F_{q} [μ^{N}] (m)$ , and thus 0_tqμ^P and 0_mqμ^N, i.e., μ^P (0) + t > 1 and μ^N (0) + m < -1, which is impossible, and hence μ^P (0) ≥ μ^P (x) and μ^N (0) ≤ μ^N (x) for all x ∈ X. Suppose there exists a, b ∈ X such that μ^P (a) < min {μ^P (a ∗ b) , μ^P (b)} and μ^N (a) > max {μ^N (a ∗ b) , μ^N (b)}. Then μ^P (a) + s ≤ 1 < min {μ^P (a ∗ b) , μ^P (b)} + s and μ^N (a) + n ≥ -1 > max {μ^N (a ∗ b) , μ^N (b)} + n for some s ∈ A and for some n ∈ ¬ A.It follows that (a ∗ b) _sqμ^P, b_sqμ^P and (a ∗ b) _nqμ^N, b_nqμ^N, i.e., $a * b \in F_{q} [μ^{P}] (s)$ and $b \in F_{q} [μ^{P}] (s)$ , and $a * b \in F_{q} [μ^{N}] (n)$ and $b \in F_{q} [μ^{N}] (n)$ . Since $F_{q} [μ^{P}] (s)$ and $F_{q} [μ^{N}] (n)$ are BCK/BCI-ideal of X, then we get $a \in F_{q} [μ^{P}] (s)$ and $a \in F_{q} [μ^{N}] (n)$ , and so a_sqμ^P and a_nqμ^N or equivalently μ^P (a) + s > 1 and μ^N (a) + n < -1. This is a contradiction. Therefore, $μ^{P} (x) \geq min {μ^{P} (x * y), μ^{P} (y)}$ $μ^{N} (x) \leq max {μ^{N} (x * y), μ^{N} (y)}$ for all x, y ∈ X. Hence μ is a bipolar fuzzy BCK/BCI-ideal of X.□

Theorem 4.5. Let μ = (μ^P, μ^N) be a bipolar fuzzy set of X and $(F [μ], A)$ be a bipolar ∈-fuzzy soft set over X with A = (0, 0.5] and ¬A = [-0.5, 0). Then the following assertions are equivalent

(i) μ is an (∈ , ∈ ∨ q)-bipolar fuzzy ideal of X

(ii) $(F [μ], A)$ is a bipolar fuzzy soft ideal of X.

Proof: Assume that μ is an (∈ , ∈ ∨ q)-bipolar fuzzy ideal of X. Let t ∈ A and m ∈ ¬ A. By using Proposition 2.10 (i), we get μ^P (0) ≥ min {μ^P (x) , 0.5} and μ^N (0) ≤ max {μ^N (x) , -0.5} for all x ∈ F [μ] (t, m). Then it follows that μ^P (0) ≥ min {μ^P (x) , 0.5} ≥ min {t, 0.5} = t, this imply 0_t ∈ μ^P and μ^N (0) ≤ max {μ^N (x) , -0.5} ≤ max {m, - 0.5} = m, this imply 0_m ∈ μ^N. Hence, we get $0 \in F [μ^{P}] (t)$ and $0 \in F [μ^{N}] (m)$ , i.e., 0 ∈ F [μ] (t, m). Let x, y ∈ X be such that $x * y \in F [μ^{P}] (t)$ and $y \in F [μ^{P}] (t)$ , and also $x * y \in F [μ^{N}] (m)$ and $y \in F [μ^{N}] (m)$ . Then (x ∗ y) _t ∈ μ^P and y_t ∈ μ^P, and also (x ∗ y) _m ∈ μ^N and y_m ∈ μ^N, or equivalently, μ^P (x ∗ y) ≥ t and μ^P (y) ≥ t, and also μ^N (x ∗ y) ≤ m and μ^N (y) ≤ m. Then by Proposition 2.10 (ii) and (iii), we have μ^P (x) ≥ min {μ^P (x ∗ y) , μ^P (y) , 0.5} ≥ min {t, 0.5} = t, thisimply x_t ∈ μ^P, and also, μ^N (x) ≤ max {μ^N (x ∗ y) , μ^N (y) , -0.5} ≤ max {m, - 0.5} = m, this imply x_m ∈ μ^N. Hence, x ∈ F [μ^P, μ^N] (t, m), and so $(F [μ], A)$ is a bipolar fuzzy soft ideal over X. Conversely, suppose that (i) and (ii) are valid. If there a ∈ X such that μ^P (0) < min {μ^P (a) , 0.5} and μ^N (0) > max {μ^N (a) , -0.5}, then μ^P (0) < t ≤ max {μ^P (a) , 0.5} and μ^N (0) > m ≥ max {μ^N (a) , -0.5} for some t ∈ A and for some m ∈ ¬ A. Then it follows that $0_{t} \bar{\in} μ^{P}$ and $0_{m} \bar{\in} μ^{N}$ , i.e., $0 \notin F [μ^{P}] (t)$ and $0 \notin F [μ^{N}] (m)$ , or equivalently, 0 ∉ F [μ] (t, m), a contradiction. Hence, μ^P (0) ≥ min {μ^P (x) , 0.5} and μ^N (0) ≤ max {μ^N (x) , -0.5} for all x ∈ X. Assume that if there exist a′, b′ ∈ X such that μ^P (a′) < min {μ^P (a′ ∗ b′) , μ^P (b′) , 0.5} and μ^N (a′) > max {μ^N (a′ ∗ b′) , μ^N (b′) , -0.5}. Taking

$t_{0} = (\frac{1}{2} {μ^{P} (a^{'})} + min {μ^{P} (a^{'} * b^{'}), μ^{P} (b^{'}), 0.5})$ $m_{0} = (\frac{1}{2} {μ^{N} (a^{'})} + max {μ^{N} (a^{'} * b^{'}), μ^{N} (b^{'}), - 0.5}) .$

We have t₀ ∈ A and m₀ ∈ ¬ A such that μ^P (a′) < t₀ < min {μ^P (a′ ∗ b′) , μ^P (b′) , 0.5} and μ^N (a′) > m₀ > max {μ^N (a′ ∗ b′) , μ^N (b′) , -0.5}. Hence, we get (a′ ∗ b′) _{t
₀} ∈ μ^P and $b_{t_{0}}^{'} \in μ^{P}$ but $a_{t_{0}}^{'} \bar{\in} μ^{P}$ and (a′ ∗ b′) _{m
₀} ∈ μ^N and $b_{m_{0}}^{'} \in μ^{N}$ but $a_{m_{0}}^{'} \bar{\in} μ^{N}$ . These implies that $a^{'} * b^{'} \in F [μ^{P}] (t_{0})$ and $b^{'} \in F [μ^{P}] (t_{0})$ but $a^{'} \notin F [μ^{P}] (t_{0})$ , and $a^{'} * b^{'} \in F [μ^{N}] (m_{0})$ , $b^{'} \in F [μ^{N}] (m_{0})$ but $a^{'} \notin F [μ^{N}] (m_{0})$ , i.e., a′ ∉ F [μ] (t₀, m₀), is a contradiction. Hence, μ^P (x) ≥ min {μ^P (x ∗ y) , μ^P (y) , 0.5} and μ^N (x) ≤ max {μ^N (x ∗ y) , μ^N (y) , -0.5} for all x, y ∈ X. Therefore, μ is an (∈ , ∈ ∨ q)-bipolar fuzzy ideal of X.□

Example 4.6. Consider a BCK/BCI-algebra X = {0, a, b, c, d} with the following Caley table $\begin{matrix} * & 0 a b c d \\ 0 & 00000 \\ a & a 0 a 0 a \\ b & b b 0 b 0 \\ c & c a c 0 c \\ d & d d d d 0 \end{matrix}$ we define the bipolar fuzzy set μ of X as follows μ^P (0) =0.7, μ^P (a) = μ^N (b) =0.3, μ^P (b) = μ^P (d) =0.2 and μ^N (0) = -0.9, μ^N (a) = -0.6, μ^N (b) = -0.4, μ^N (c) = -0.7 and μ^N (d) = -0.3 is an (∈ , ∈ ∨ q)-bipolar fuzzy ideal of X. Let $(F [μ], A)$ be a bipolar ∈-fuzzy soft set over X with A = (0, 0.5] and ¬A = [-0.5, 0). Then $F [μ^{P}] (t) = {\begin{matrix} X, & if t \in (0, 0.2] \\ {0, a, c}, & if t \in (0.2, 0.5] . \end{matrix}$ and $F [μ^{N}] (s) = {\begin{matrix} X, & if s \in [- 0.3, 0) \\ {0, a, b, c}, & if s \in (- 0.3, - 0.5] . \end{matrix}$ which is a bipolar fuzzy ideals of X. Hence, $(\tilde{F} [μ^{N}], A)$ is a bipolar fuzzy soft ideal over X.

Theorem 4.7. Let μ = (μ^P, μ^N) be a bipolar fuzzy set of X and let $(F [μ], A)$ be a bipolar ∈-fuzzy soft set over X with A = (0.5, 1] and ¬A = [-1, - 0.5). Then $(F [μ], A)$ is a bipolar fuzzy soft ideal over X if and only if μ satisfies the following conditions

(i) max {μ^P (0) , 0.5} ≥ μ^P (x) and min {μ^N (0) , -0.5} ≤ μ^N (x),

(ii) max {μ^P (x) , 0.5} ≥ min {μ^P (x ∗ y) , μ^P (y)},

(iii) min {μ^N (x) , -0.5} ≤ max {μ^N (x ∗ y) , μ^N (y)} for all x, y ∈ X.

Proof: Assume that $(F [μ], A)$ is a soft ideal over X. If there is an element a ∈ X such that the condition (i) is not valid, then μ^P (a) ∈ A, μ^N (a) ∈ ¬ A and a ∈ F [μ^P (a)] and a ∈ F [μ^N (a)] but μ^P (0) < μ^P (a) and μ^N (0) > μ^N (a) which implies that 0 ∉ F [μ], which is a contradiction. Hence, (i) is valid. Let us now take $max {μ^{P} (a), 0.5} < min {μ^{P} (a * b), μ^{P} (b)} = t$ $min {μ^{N} (a), - 0.5} > max {μ^{N} (a * b), μ^{N} (b)} = m$ for some a, b ∈ X. Then t ∈ A and $a * b, b \in F [μ^{P}] (t)$ , and m ∈ ¬ A and $a * b, b \in F [μ^{N}] (m)$ . But we get $a \notin F [μ^{P}] (t)$ as μ^P (a) < t and $a \notin F [μ^{N}]$ as μ^N (a) > m, which is a contradiction, so (ii) and (iii) are valid. Conversely, μ satisfies (i), (ii) and (iii). Let t ∈ A and m ∈ ¬ A. For any x ∈ F [μ], we get $max {μ^{P} (0), 0.5} \geq μ^{P} (x) \geq t > 0.5$ $min {μ^{N} (0), - 0.5} \leq μ^{N} (x) \leq m < - 0.5$ and so, μ^P (0) ≥ t and μ^N (0) ≤ m. These implies that 0_t ∈ μ^P and 0_m ∈ μ^N. Hence, $0 \in F [μ^{P}] (t)$ and $0 \in F [μ^{N}] (m)$ i.e., 0 ∈ F [μ] (t, m). Let x, y ∈ X be such that $x * y \in F [μ^{P}] (t)$ and $y \in F [μ^{P}] (t)$ , and $x * y \in F [μ^{N}] (m)$ and $y \in F [μ^{N}] (m)$ . Then (x ∗ y) _t ∈ μ^P and y_t ∈ μ^P which imply μ^P (x ∗ y) ≥ t and μ^P (y) ≥ t, and (x ∗ y) _m ∈ μ^N and y_m ∈ μ^N which imply μ^N (x ∗ y) ≤ m and μ^N (y) ≤ m. Hence,

$max {μ^{P} (x), 0.5} \geq min {μ^{P} (x * y), μ^{P} (y)} \geq t > 0.5$ $min {μ^{N} (x), - 0.5} \leq max {μ^{N} (x * y), μ^{N} (y)} \leq m < - 0.5$ which indicate that μ^P (x) ≥ t, i.e., x_t ∈ μ^P and μ^N (x) ≤ m, i.e., x_m ∈ μ^N. Therefore, x ∈ F [μ] (t, m) and hence, $(F [μ], A)$ is a bipolar fuzzy soft ideal over X.□

5 Conclusions

In this paper, we introduced the notion of ∈-soft set and q-soft set based on bipolar fuzzy set, and gave a characterizations for an ∈-soft set and a q-soft set to be bipolar fuzzy soft BCK/BCI-algebra. We also introduced the notion of (∈ , ∈ ∨ q)-bipolar fuzzy BCK/BCI-subalgebra/ ideal. We characterized the relation between (∈ , ∈ ∨ q)-bipolar fuzzy subalgebras/ideals with bipolar fuzzy subalgebras/ideals of BCK/BCI-algebra, we provided the characterizations for an ∈-soft and a q-soft to be bipolar fuzzy soft BCK/BCI-algebra.

In our future study of bipolar fuzzy structure of BCK/BCI-algebra, may be considered with the following topics: (i) bipolar (T, S)-fuzzy soft BCK/BCI-algebra, where T and S are triangular norm and conorm respectively, (ii) bipolar $(\bar{\in}, \bar{\in} \lor \bar{q})$ -fuzzy soft BCK/BCI-algebra, (iii) (∈ , ∈ ∨ q)-bipolar fuzzy soft (p-, a- and q-)ideals and their relations.

Footnotes

Acknowledgements

The authors are highly grateful to referees and Professor Violeta Fotea, Associate Editor, for their valuable comments and suggestions for improving the paper.

References

Molodtsov

, Soft set theory-first results, Comput Math Appl 37 (1999), 19–31.

Molodtsov

, The Theory of Soft Sets (in Russian), URSS Publishers, Moscow, 2004.

Maji

P.K.

, Biswas

and Roy

A.R.

, Soft set theory, Comput Math Appl 45 (2003), 555–562.

Maji

P.K.

, Roy

A.R.

and Biswas

, An application of soft sets in a decision making problem, Comput Math Appl 44 (2002), 1077–1083.

Dubois

, Kaci

and Prade

, Bipolarity in Reasoning and Decision, an Introduction, Int Con on Inf Pro Man Unc IPMU’04, 2004, pp.959–966.

Zhang

W.R.

, Bipolar fuzzy sets and relations: A computational framework for cognitive and modeling and multiagent decision analysis, Proc of IEEE conf (1994), 305–309.

Zhang

W.R.

, Bipolar fuzzy sets, Proc of FUZZ-IEEE (1998), 835–840.

Zhang

W.R.

and Zhang

, YinYang bipolar logic and bipolar fuzzy logic, Inform Sci 165 (2004), 265–287.

Zadeh

L.A.

, Fuzzy sets, Inform Control 8 (1965), 338–353.

10.

Lee

K.M.

, Bipolar-valued fuzzy sets and their basic operations, Proc Int Conf, Bangkok, Thailand, 2000, pp.307–317.

11.

Lee

K.M.

, Comparison of interval-valued fuzzy sets, intuitionistic fuzzy sets, and bipolar-valued fuzzy sets, J Fuzzy Logic Intell Sys 14 (2004), 125–129.

12.

Imai

and Iseki

, On axiom system of propositional calculi, XIV Proc Japan Academy 42 (1966), 19–22.

13.

Iseki

, An algebra related with a propositional calculus, Proceedings of the Japan Academy 42 (1966), 26–29.

14.

Jun

Y.B.

and Shim

W.H.

, Fuzzy strong implicative hyper BCK-ideals of hyper BCK-algebras, Inform Sci 170 (2005), 351–361.

15.

Jun

Y.B.

and Xin

X.L.

, Involutory and invertible fuzzy BCK-algebras, Fuzzy Sets and Systems 117 (2001), 463–469.

16.

Liu

Y.L.

, Xu

and Meng

, BCI-implicative ideals of BCIalgebras, Inform Sci 177 (2007), 4987–4996.

17.

Lee

K.J.

and Jun

Y.B.

, Bipolar fuzzy a-ideals of BCIalgebras, Commun Korean Math Soc 26(4) (2011), 531–542.

18.

Meng

and Jun

Y.B.

, BCK-algebras, Kyungmoon Sa Co, Seoul, 1994.

19.

Meng

and Guo

, On fuzzy ideals inBCK/BCI-algebras, Fuzzy Sets and Systems 149(3) (2005), 509–525.

20.

Muhiuddin

, Kim

H.S.

, Song

S.Z.

and Jun

Y.B.

, Hesitant fuzzy translations and extensions of subalgebras and ideals in BCK/BCI-algebras, J Int Fuzzy syst 32(1) (2017), 43–48.

21.

Jana

and Pal

, On (α, β)-Union-soft sets in BCK/BCIalgebras, Mathematics 7 (2019), 252.

22.

Senapati

and Shum

K.P.

, Atanassov’s intuitionistic fuzzy binormed KU-subalgebras of a KU-algebra, Missouri J Math Sci 29 (2017), 92–112.

23.

, Zhan

and Jun

Y.B.

, Some kinds of (∈_γ, ∈ _γ ∨ qδ)-fuzzy ideals of BCI-algebras, Comput Math Appl 61(4) (2011), 1005–1015.

24.

, Zhan

and Jun

Y.B.

, New types of fuzzy ideals of BCI-algebras, Neural Comput Applic 21 (2012), S19–S27.

25.

Jana

, Senapati

, Bhowmik

and Pal

, On intuitionistic fuzzy G-subalgebras of G-algebras, Fuzzy Inf Eng 7(2) (2015), 195–209.

26.

Jana

, Pal

, Senapati

and Bhowmik

, Atanassov’s intutionistic L-fuzzy G-subalgebras of G-algebras, J Fuzzy Math 23(2) (2015), 195–209.

27.

Jana

and Senapati

, Cubic G-subalgebras of G-algebras, Annals of Pure and Applied Mathematics 10(1) (2015), 105–115.

28.

Jana

and Pal

, Generalized intuitionistic fuzzy ideals of BCK/BCI-algebras based on 3-valued logic and its computational study, Fuzzy Inf Eng 9 (2017), 455–478.

29.

Jana

, Senapati

and Pal

, Derivation, f-derivation and generalized derivation of KUS-algebras, Cogent Mathematics 2 (2015), 1–12.

30.

Jana

and Pal

, Applications of new soft intersection set on groups, Ann Fuzzy Math Inform 11(6) (2016), 923–944.

31.

Jana

, Senapati

and Pal

, (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebras of BCI-algebra, J Int Fuzzy Syst 31 (2016), 613–621.

32.

Jana

and Pal

, Applications of (α, β)-soft intersectional sets on BCK/BCI-algebras, Int J of Intelligent Systems Technologies and Applications 16(3) (2017), 269–288.

33.

Jana

, Senapati

and Pal

, On t-Derivations of complicaated subtraction algebraas, Journal of Discrete Mathematical Sciences and Cryptography 20(8) (2018), 1583–1595.

34.

Jana

, Pal

and Saied

A.B.

, (∈, ∈ ∨ q)-bipolar fuzzy BCK/BCI-algebras, Missouri J of Math Sci 29(2) (2017), 139–160.

35.

Jana

and Pal

On (∈_α, ∈ _β ∨ q_β)-fuzzy soft BCI algebras, Missouri J of Math Sci 29(2) (2017), 197–215.

36.

Bej

and Pal

, Doubt Atanassovs intuitionistic fuzzy Subimplicative ideals in BCI-algebras, Int J Comput Int Sys 8(2) (2015), 240–249.

37.

Murali

, Fuzzy points of equivalent fuzzy subsets, Inform Sci 158 (2004), 277–288.

38.

P.M.

and Liu

Y.M.

, Fuzzy topology I, Neighborhood structure of a fuzzy point and Moore-Smith convergence, J Math Anal Appl 76 (1980), 571–599.

39.

Bhakat

S.K.

and Das

, On the definition of a fuzzy subgroup, Fuzzy Sets and Systems 51 (1992), 235–241.

40.

Bhakat

S.K.

and Das

, (∈, ∈ ∨ q)-fuzzy subgroup, Fuzzy Sets and Systems 80 (1996), 359–368.

41.

Rosenfeld

, Fuzzy groups, J Math Anal Appl 35 (1971)512–517.

42.

Chen

, Tsang

E.C.C.

, Yeung

D.S.

and Wang

, The parametrization reduction of soft sets and its applications, Comput Math Appl 49 (2005), 757–763.

43.

Feng

, Jun

Y.B.

and Zhao

, Soft semirings, Comput Math Appl 56 (2008), 2621–2628.

44.

Jiang

, Tang

, Chen

, Liu

and Tang

, Intervalvalued intuitionistic fuzzy soft sets and their properties, Comput Math Appl 60 (2010), 906–918.

45.

Jun

Y.B.

, Öztürk

M.A.

and Muhiuddin

, A generalization of (∈, ∈ ∨ q)-fuzzy subgroups, International Journal of Algebra and Statistics 5(1) (2016), 7–18.

46.

Zhan

, Jun

Y.B.

and Davvaz

, On (∈, ∈ ∨ q)-fuzzy ideals of BCI-algebras, Iran J Fuzzy Syst 6 (2009), 81–94.

47.

Zhang

, Zhang

, Liu

and Gang

, New kinds of fuzzy ideals in BCI-algebras, Fuzzy Optim Decis Mak 5 (2006), 177–186.

48.

Jun

Y.B.

, On (α, β)-fuzzy subalgebras of BCK/BCIalgebras, Bull Korean Math Soc 42(4) (2005), 703–711.

49.

Jun

Y.B.

, On (α, β)-fuzzy ideals of BCK/BCI-algebrs, Sci Math Jpn (2004), 101–105.

50.

, Zhan

J.M.

, Davvaz

and Jun

Y.B.

, Some kinds of (∈, ∈ ∨ q)-interval-valued fuzzy ideals of BCI-algebras, Inform Sci 178 (2008), 3738–3754.

51.

Lee

K.J.

, Bipolar fuzzy subalgebras and bipolar fuzzy ideals of BCK/BCI-algebras, Bull Malays Math Sci Soc 32(3) (2009), 361–373.

52.

Jun

Y.B.

, Soft BCK/BCI-algebras, Comput Math Appl 56 (2008), 1408–1413.

53.

Jun

Y.B.

and Park

C.H.

, Applications of soft sets in ideal theory of BCK/BCI-algebras, Inform Sci 178 (2008), 2466–2475.

54.

Jun

Y.B.

, Lee

K.J.

and Park

C.H.

, Soft set theory applied to ideals in d-algebras, Comput Math Appl 57 (2009), 367–378.

55.

Jun

Y.B.

, Lee

K.J.

and Zhan

, Soft p-ideals of soft BCIalgerbas, Comput Math Appl 58 (2009), 2060–2068.

56.

Aktas

and Cagman

, Soft sets and soft groups, Inform Sci 177 (2007), 2726–2735.

57.

Huang

Y.S.

, BCI-algebra, Science Press, Beijing, 2006.