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

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 ) = { t ∈ ( 0 , 1 ] if y = x 0 if y ≠ x . 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 _t Φμ, where Φ ∈ {∈ , q, ∈ ∨ q, ∧ q}. To say that x _t  ∈ μ (respectively, x _t qμ) 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 _t qμ (respectively, x _t  ∈ μ and x _t qμ). To say that x t Φ ¯ μ means that x _t Φμ 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 ∪ B ) ( x ) = { max { μ A P ( x ) , μ B P ( x ) } , min { μ A N ( x ) , μ B N ( x ) } } ( A ∩ 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 ∈ μ A P , y s ∈ μ A P ⇒ ( x ∗ y ) min ( t , s ) ∈ μ A P and x m ∈ μ A N , y n ∈ μ A N ⇒ ( x ∗ y ) max ( m , n ) ∈ 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 ) ≥ min { μ A P ( x ) , μ A P ( y ) , 0.5 } and μ A N ( x ∗ y ) ≤ 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 ∈ μ A P ⇒ 0 t ∈ μ A P and x m ∈ μ A N ⇒ 0 m ∈ μ A N , for all x ∈ X, t ∈ [0, 1], m ∈ [-1, 0]) (ii) ( x ∗ y ) t ∈ μ A P , y s ∈ μ A P ⇒ x min ( t , s ) ∈ μ A P , for all x, y ∈ X, t, s ∈ [0, 1] (iii) ( x ∗ y ) m ∈ μ A N , y n ∈ μ A N ⇒ x max ( m , n ) ∈ 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 ) ≥ min { μ A P ( x ) , 0.5 } and μ A N ( 0 ) ≤ max { μ A N ( x ) , - 0.5 } for all x ∈ X (ii) μ A P ( x ) ≥ min { μ A P ( x ∗ y ) , μ A P ( y ) , 0.5 } for all x, y ∈ X (iii) μ A N ( x ) ≤ 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 → 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 _t qμ} 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 ( ∈ ) = { x , μ A ( ∈ ) P , μ A ( ¬ ∈ ) N } where μ A ( ∈ ) P : U → ( 0 , 1 ] and μ A ( ¬ ∈ ) N : U → [ - 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 ( ∈ ) P denotes the degree of x keeping the parameter ∈, μ A ( ∈ ) 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 ) = { t ∈ ( 0 , 1 ] if y = x 0 if y ≠ x .

μ N ( y ) = { m ∈ [ - 1 , 0 ) if y = x 0 if y ≠ x . 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 _t Φμ ^P , x _m Φμ ^N ), where Φ ∈ {∈ , q, ∈ ∨ q, ∈ ∧ q} . To say that x _t  ∈ μ ^P (respectively, x _t qμ ^P ) and x _m  ∈ μ ^N (respectively, x _m qμ ^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 _t qμ ^P (respectively, x _t  ∈ μ ^P and x _t qμ ^P ) and x _m  ∈ μ ^N or x _m qμ ^N (respectively, x _m  ∈ μ ^N and x _m qμ ^N ). To say that ( x t Φ ¯ μ P , x m Φ ¯ μ N ) means that x _t Φμ ^P does not hold and x _m Φμ ^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: ∗ 0 a 1 2 3 0 0 0 3 2 1 a a 0 3 2 1 1 1 1 0 3 2 2 2 2 1 0 3 3 3 3 2 1 0 and ( F [ μ ] , A ) be a bipolar fuzzy soft set over X, where A = (0, 1] and ¬A = [-1, 0) and F : A → P ( X ) and F : ¬ A → P ( X ) are a set-valued functions defined by F [ μ P ] ( t ) = { ∅ , if 0.8 < t { 0 } , if 0.6 < t ≤ 0.8 { 0 , a } , if 0.3 < t ≤ 0.6 X , if 0 ≤ t ≤ 0.3 . and F [ μ N ] ( s ) = { ∅ , if - 1 ≤ s < - 0.8 { 0 } , if - 0.7 ≤ s < - 0.5 { 0 , a } , if - 0.5 ≤ s < - 0.2 X , if - 0.2 ≤ s < 0 . 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: ∗ 0 a b c d 0 0 0 0 0 0 a a 0 a 0 0 b b b 0 0 0 c c c c 0 0 d d c d a 0 Let us define the set-valued functions such that F [ μ P ] ( t ) = { ∅ , if 0.7 < t { 0 , b } , if 0.2 < t ≤ 0.7 X , if 0 ≤ t ≤ 0.2 . and F [ μ N ] ( s ) = { ∅ , if - 1 < s < - 0.8 { 0 , b } , if - 0.8 ≤ s < - 0.7 X , if - 0.7 ≤ s < 0 .

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 F [ μ P ] : A → P ( X ) , t ↦ { x ∈ X | x t ∈ μ P } , F q [ μ P ] : A → P ( X ) , t ↦ { x ∈ X | x t q μ P } , F [ μ N ] : ¬ A → P ( X ) , m ↦ { x ∈ X | x m ∈ μ N } and F q [ μ N ] : ¬ A → P ( X ) , m ↦ { x ∈ X | x m q μ N } . 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 ∈ F [ μ P ] ( t ) and a , b ∈ F [ μ N ] ( m ) . But a ∗ b ∉ F [ μ P ] ( t ) and a ∗ b ∉ 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 ∈ F [ μ P ] ( t ) , and x , y ∈ 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 ∈ F [ μ P ] ( t ) and x ∗ y ∈ 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 _t qμ ^P , y _t qμ ^P and x _m qμ ^N , y _m qμ ^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)  _t qμ ^P and (x ∗ y)  _m qμ ^N , i.e., x ∗ y ∈ F q [ μ P ] ( t ) and x ∗ y ∈ 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 _t qμ ^P , b _t qμ ^P and a _m qμ ^N , b _m qμ ^N , but ( a ∗ b ) t q ¯ μ P and ( a ∗ b ) m q ¯ μ N i.e., a t ∈ F q [ μ P ] ( t ) , b t ∈ F q [ μ P ] ( t ) and a m ∈ F q [ μ N ] ( m ) , b m ∈ F q [ μ N ] ( m ) , but a ∗ b ∉ F q [ μ P ] ( t ) and a ∗ b ∉ 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 ∈ ¯ μ P and ( a ∗ b ) m ∈ ¯ μ N . It follows that a , b ∈ F [ μ P ] ( t ) and a , b ∈ F [ μ N ] ( m ) but a ∗ b ∉ F [ μ P ] ( t ) and a ∗ b ∉ 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 ∈ F [ μ P ] ( t ) , and x , y ∈ 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 ∈ F [ μ P ] ( t ) and x ∗ y ∈ 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 ∈ F [ μ P ] ( t ) and x , y ∈ 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 μ P ( x ∗ y ) ≥ min { μ P ( x ) , μ P ( y ) , 0.5 } ≥ min { t , 0.5 } = t μ N ( x ∗ y ) ≤ max { μ N ( x ) , μ N ( y ) , - 0.5 } ≤ min { m , - 0.5 } = m .

So that (x ∗ y)  _t  ∈ μ ^P and (x ∗ y)  _m  ∈ μ ^N or equivalently x ∗ y ∈ F [ μ P ] ( t ) and x ∗ y ∈ 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 ≤ min { μ P ( a ) , μ P ( b ) , 0.5 } μ N ( a ∗ b ) > m ≥ 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 ∈ F [ μ P ] ( t ) , b ∈ F [ μ P ] ( t ) and a ∈ F [ μ N ] ( m ) , b ∈ F [ μ N ] ( m ) . Since F [μ] is a subalgebras of X, it follows that a ∗ b ∈ F [ μ P ] ( t ) for all t ≤ 0.5 and a ∗ b ∈ 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

∗ 0 a b c 0 0 a b c a a 0 c b b b c 0 a c c b a 0 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 ) = { X , if t ∈ ( 0 , 0.3 ] { 0 , a } , if t ∈ ( 0.3 , 0.5 ] . F [ μ N ] ( s ) = { X , if s ∈ [ - 0.3 , 0 ) { 0 , b } , if s ∈ ( - 0.3 , - 0.5 ] . 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: ∗ 0 a b c d 0 0 0 0 0 0 a a 0 a 0 a b b b 0 b 0 c c a c 0 c d d d d d 0 Let us define the set-valued functions such that F [ μ P ] ( t ) = { ∅ , if 0.7 < t { 0 } , if 0.3 < t ≤ 0.7 { 0 , a , c } , if 0.2 < t ≤ 0.3 X , if 0 ≤ t ≤ 0.2 . and F [ μ N ] ( s ) = { ∅ , if - 1 ≤ s < - 0.9 { 0 } , if - 0.9 ≤ s < - 0.7 { 0 , c } , if - 0.7 ≤ s < - 0.6 { 0 , a , c } , if - 0.6 ≤ s < - 0.4 { 0 , a , b , c } , if - 0.4 ≤ s < - 0.3 X , if - 0.3 ≤ s < 0 .

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 ∈ F [ μ P ] ( t ) and 0 ∈ F [ μ N ] ( m ) . Let x, y ∈ X be such that ( x ∗ y ) ∈ F [ μ P ] ( t ) and y ∈ F [ μ P ] ( t ) and also, ( x ∗ y ) ∈ F [ μ N ] ( m ) and y ∈ 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 ∈ F [ μ P ] ( t ) and x ∈ 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 ∉ F [ μ P ] ( t ) and 0 ∉ 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 ∈ F [ μ P ] ( s ) and b ∈ F [ μ P ] ( s ) but a s ∉ F [ μ P ] ( s ) and a ∗ b ∈ F [ μ N ] ( n ) , b ∈ F [ μ N ] ( n ) but a ∉ F [ μ N ] ( n ) . This is a contradiction, so μ P ( x ) ≥ min { μ P ( x ∗ y ) , μ P ( y ) } μ N ( x ) ≤ 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 ) ≠ ∅ and F q [ μ N ] ( m ) ≠ ∅ . If 0 ∉ F q [ μ P ] ( t ) and 0 ∉ F q [ μ N ] ( m ) , then 0 t q ¯ μ P and x m 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 ∈ F q [ μ P ] ( t ) and 0 ∈ F q [ μ N ] ( m ) , i.e, 0 ∈ F _q  [μ] (t, m). Let x, y ∈ X be such that x ∗ y ∈ F q [ μ P ] ( t ) , y ∈ F q [ μ P ] ( t ) and x ∗ y ∈ F q [ μ N ] ( m ) , y ∈ F q [ μ N ] ( m ) . Then (x ∗ y)  _t qμ ^P , y _t qμ ^P and (x ∗ y)  _m qμ ^N , y _m qμ ^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 _t qμ ^P and x _m qμ ^N , i.e., x ∈ F q [ μ P ] ( t ) and x ∈ 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 _t qμ ^P and a _m qμ ^N , and so F q [ μ P ] ( t ) ≠ ∅ and F q [ μ N ] ( m ) ≠ ∅ , i.e., F _q  [μ] (t, m)≠ ∅. Hence 0 ∈ F q [ μ P ] ( t ) , and 0 ∈ F q [ μ N ] ( m ) , and thus 0 _t qμ ^P and 0 _m qμ ^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)  _s qμ ^P , b _s qμ ^P and (a ∗ b)  _n qμ ^N , b _n qμ ^N , i.e., a ∗ b ∈ F q [ μ P ] ( s ) and b ∈ F q [ μ P ] ( s ) , and a ∗ b ∈ F q [ μ N ] ( n ) and b ∈ 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 ∈ F q [ μ P ] ( s ) and a ∈ F q [ μ N ] ( n ) , and so a _s qμ ^P and a _n qμ ^N or equivalently μ ^P  (a) + s > 1 and μ ^N  (a) + n < -1. This is a contradiction. Therefore, μ P ( x ) ≥ min { μ P ( x ∗ y ) , μ P ( y ) } μ N ( x ) ≤ 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 ∈ F [ μ P ] ( t ) and 0 ∈ F [ μ N ] ( m ) , i.e., 0 ∈ F [μ] (t, m). Let x, y ∈ X be such that x ∗ y ∈ F [ μ P ] ( t ) and y ∈ F [ μ P ] ( t ) , and also x ∗ y ∈ F [ μ N ] ( m ) and y ∈ 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 ∈ ¯ μ P and 0 m ∈ ¯ μ N , i.e., 0 ∉ F [ μ P ] ( t ) and 0 ∉ 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 = ( 1 2 { μ P ( a ′ ) } + min { μ P ( a ′ ∗ b ′ ) , μ P ( b ′ ) , 0.5 } ) m 0 = ( 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 ′ ∈ μ P but a t 0 ′ ∈ ¯ μ P and (a′ ∗ b′)  _{m 0}  ∈ μ ^N and b m 0 ′ ∈ μ N but a m 0 ′ ∈ ¯ μ N . These implies that a ′ ∗ b ′ ∈ F [ μ P ] ( t 0 ) and b ′ ∈ F [ μ P ] ( t 0 ) but a ′ ∉ F [ μ P ] ( t 0 ) , and a ′ ∗ b ′ ∈ F [ μ N ] ( m 0 ) , b ′ ∈ F [ μ N ] ( m 0 ) but a ′ ∉ 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 ∗ 0 a b c d 0 0 0 0 0 0 a a 0 a 0 a b b b 0 b 0 c c a c 0 c d d d d d 0 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 ) = { X , if t ∈ ( 0 , 0.2 ] { 0 , a , c } , if t ∈ ( 0.2 , 0.5 ] . and F [ μ N ] ( s ) = { X , if s ∈ [ - 0.3 , 0 ) { 0 , a , b , c } , if s ∈ ( - 0.3 , - 0.5 ] . which is a bipolar fuzzy ideals of X. Hence, ( F ˜ [ μ N ] , A ) is a bipolar fuzzy soft ideal over X.