Fuzzy soft positive implicative hyper BCK -ideals in hyper BCK -algebras

Abstract

Fuzzy soft set theory is applied to hyper BCK-algebras. The notion of fuzzy soft positive implicative hyper BCK-ideals is introduced, and several properties are investigated. The relation between fuzzy soft positive implicative hyper BCK-ideal and fuzzy soft hyper BCK-ideal is considered. Characterizations of fuzzy soft positive implicative hyper BCK-ideal are provided. Using the notion of positive implicative hyper BCK-ideal, a fuzzy soft weak (strong) hyper BCK-ideal is established.

1. Introduction

Hyperstructure theory was born in 1934 when Marty defined hypergroups, began to analyze their properties, and applied them to groups and relational algebraic functions (see [22]). Since then, many papers and several books have been written on this topic. Nowadays, hyperstructures have a lot of applications in several branches of mathematics and computer sciences (see [1–3 , 24–28]). In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. In [18], Jun et al. applied the hyperstructures to BCK-algebras, and introduced the concept of a hyper BCK-algebra which is a generalization of a BCK-algebra. Sine then, Jun et al. studied more notions and results in [12 , 17]. Borzooei et al. [6] introduced the concept of the hyper K-algebra which is a generalization of the hyper BCK-algebra, and Zahedi et al. [29] defined the notions of (weak, strong) implicative hyper K-algebras. Borumand Saeidet al. [5] studied (weak) implicative hyper K-ideals in hyper K-algebras. Also, several fuzzy versions of hyper BCK-algebras have been considered in [13, 15]. Molodtsov [23] proposed a new approach, which was called soft set theory, for modeling uncertainty. In [10], Jun applied the notion of soft sets to the theory of BCK/BCI-algebras, and Jun et al. [12] studied ideal theory of BCK/BCI-algebras based on soft set theory. Maji et al. [21] extended the study of soft sets to fuzzy soft sets. They introduced the concept of fuzzy soft sets as a generalization of the standard soft sets, and presented an application of fuzzy soft sets in a decision making problem. Jun et al. [11] applied fuzzy soft set to BCK/BCI-algebras.

In this paper we apply the notion of fuzzy soft sets by Maji et al. to the theory of hyper BCK-algebras. We introduce the notion of fuzzy soft positive implicative hyper BCK-ideal, and investigate several properties. We discuss the relation between fuzzy soft positive implicative hyper BCK-ideal and fuzzy soft hyper BCK-ideal, and provide characterizations of fuzzy soft positive implicative hyper BCK-ideal. Using the notion of positive implicative hyper BCK-ideal, we establish a fuzzy soft weak (strong) hyper BCK-ideal.

2. Preliminaries

Let H be a nonempty set endowed with a hyper operation “cc”, that is, cc is a function from H × H to $P^{*} (H) = P (H) \ {\emptyset}$ . For $A, B \in P^{*} (H)$ , denote by AccB the set ∪ {accb ∣ a ∈ A, b ∈ B} . We shall use xccy instead of xcc {y}, {x} ccy, or {x} cc {y} .

By a BCK -algebra (see [18]) we mean a nonempty set H endowed with a hyper operation “cc” and a constant 0 satisfying the following axioms:

(xccz) cc (yccz) ⪡ xccy,

(xccy) ccz = (xccz) ccy,

xccH ⪡ {x},

x ⪡ y and y ⪡ x imply x = y,

for all x, y, z ∈ H, where x ⪡ y is defined by 0 ∈ xccy and for every A, B ⊆ H, A ⪡ B is defined by ∀a ∈ A, ∃b ∈ B such that a ⪡ b.

In a BCK -algebra H, the condition (H3) is equivalent to the condition:

xccy ⪡ {x} for all x, y ∈ H.

In any BCK-algebra H, the following hold (see [18]): $2.1 x0 ⪡ {x}, 0x ⪡ {0}, 00 ⪡ {0},$ (2.1) $2.2 {\begin{matrix} (A B) C = (A C) B, \\ A B ⪡ A, 0 A ⪡ {0}, \end{matrix}$ (2.2) $0 ⪡ x, x ⪡ x, A ⪡ A,$ (2.3) $A \subseteq B \Rightarrow A ⪡ B,$ (2.4) $0 x = {0}, 0 A = {0},$ (2.5) $A ⪡ {0} \Rightarrow A = {0},$ (2.6) $x \in x 0,$ (2.7) $x0 = {x}, A 0 = A .$ (2.8) for all x, y, z ∈ H and for all non-empty subsets A, B and C of H.

A non-empty subset I of a BCK-algebra H is said to be S-reflexive (see [7])

$(xy) \cap I \neq \emptyset \Rightarrow x y \subseteq I$ (2.9) for all x, y ∈ H.

A subset I of a BCK-algebra H is called a BCK-ideal of H (see [18]) if it satisfies $0 \in I,$ (2.10) and $x y ⪡ I, y \in I \Rightarrow x \in I$ (2.11) for all x, y ∈ H.

A subset I of a BCK-algebra H is called a strong BCK-ideal of H (see [17]) if it satisfies (2.10) and $\begin{matrix} (x y) \cap I \neq \emptyset, y \in I \Rightarrow x \in I \end{matrix}$ (2.12) for all x, y ∈ Lom.

Recall that every strong BCK-ideal is a BCK-ideal (see [17]).

A subset I of a BCK-algebra H is called a weak BCK-ideal of H (see [18]) if it satisfies (2.10) and $\begin{matrix} x y \subseteq I, y \in I \Rightarrow x \in I \end{matrix}$ (2.13) for all x, y ∈ H.

A subset I of a BCK-algebra H is called a BCK-ideal of Lom (see [16]) if it satisfies (2.10) and $\begin{matrix} (xy) z ⪡ I, y z \subseteq I \Rightarrow x z \subseteq I \end{matrix}$ (2.14) for all x, y, z ∈ H.

Molodtsov [23] 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 the power set of U and A ⊆ E .

Definition 2.1. ([23]). A pair (λ, A) is called a soft set over U, where λ is a mapping given by $λ : A \to P (U) .$

In other words, a soft set over U is a parameterized family of subsets of the universe U . For ɛ ∈ A, λ (ɛ) may be considered as the set of ɛ-approximate elements of the soft set (λ, A) . Clearly, a soft set is not a set. For illustration, Molodtsov considered several examples in [23].

Definition 2.2. ([21]). Let U be an initial universe set and E be a set of parameters. Let $F (U)$ denote the set of all fuzzy sets in U. Then (kap, A) is called a fuzzy soft set over U where A ⊆ E and kap is a mapping given by $: A \to F (U)$ .

In general, for every parameter u in A, kapu is a fuzzy set in U and it is called fuzzy value set of parameter u. If for every u ∈ A, kapu is a crisp subset of U, then (kap, A) is degenerated to be the standard soft set. Thus, from the above definition, it is clear that fuzzy soft set is a generalization of standard soft set.

Given a fuzzy set μ in a BCK-algebra Lom and a subset T of Lom, by μ_* and μ^* we mean $\begin{matrix} μ_{*} (T) = a \inT inf μ (a) μ^{*} (T) = a \in T sup μ (a), \end{matrix}$ (2.15) respectively.

Given a fuzzy set μ in a BCK-algebra Lom, we consider the following conditions: $(\forall x, y \in) (x ⪡ y \Rightarrow μ (x) \geq μ (y)),$ 2.16 $(\forall x, y \in) (μ (x) \geq min {μ_{*} (x y), μ (y)}),$ (2.17) $(\forall x, y \in) (μ (x) \geq min {μ^{*} (x y), μ (y)}),$ (2.18) $\begin{matrix} (\forall x, y \in) (\exists a \in x y) \\ (μ (x) \geq min {μ (a), μ (y)}) . \end{matrix}$ (2.19)

Definition 2.3. ([4]). A fuzzy soft set A over a BCK-algebra Lom is called a fuzzy soft BCK-ideal based on a paramenter u ∈ A over Lom (briefly, u-fuzzy soft BCK-ideal of Lom) if the fuzzy value set kapu : Lom → [0, 1] of u satisfies two conditions (2.16) and (2.17). If A is a fuzzy soft BCK-ideal based on a paramenter u over Lom for all u ∈ A, we say that A is a fuzzy soft BCK-ideal of Lom.

Proposition 2.4. ([4]). For every fuzzy soft BCK-ideal A of Lom, the following assertions are valid.

A satisfies the condition $(\forall x \in) ((0) \geq (x)) .$ (2.20) where u is any parameter in A.

If A satisfies the following condition: (2.21) where u is any parameter in A, then $\begin{matrix} \begin{matrix} (\forall x, y \in) (\exists a \in x y) \\ ((x) \geq min {(a), (y)}) . \end{matrix} \end{matrix}$

Definition 2.5. ([4]). A fuzzy soft set A over Lom is called a fuzzy soft weak BCK-ideal based on a paramenter u ∈ A over Lom (briefly, u-fuzzy soft weak BCK-ideal of Lom) if the fuzzy value set $: \to [0, 1]$ of u satisfies conditions (2.17) and (2.20). If A is a fuzzy soft weak BCK-ideal based on u over Lom for all u ∈ A, we say that A is a fuzzy soft weak BCK-ideal of Lom.

Definition 2.6. ([4]). A fuzzy soft set A over Lom is called a fuzzy soft strong BCK-ideal over Lom based on a parameter u in A (briefly, u-fuzzy soft strong BCK-ideal of Lom) if the fuzzy value set $: \to [0, 1]$ of u satisfies the condition (2.18) and $(\forall x \in H) (\tilde{λ} {[u]}_{*} (x \circ x) \geq \tilde{λ} [u] (x)) .$ (2.22) If A is a u-fuzzy soft strong BCK-ideal of Lom for all u ∈ A, we say that A is a fuzzy soft strong BCK-ideal of Lom.

3. Fuzzy soft positive implicative hyper BCK-ideals

In what follows, let Lom and E be a BCK-algebra and a set of parameters, respectively, and A be a subset of E unless otherwise specified.

We first consider the following condition $x ⪡ y \Rightarrow (x) \geq (y)$ (3.1) for all x, y ∈ Lom. It is clear that the condition (2.20) is induced from the condition (3.1). But the converse is not true as seen in the following example.

Example 3.1. Consider a hyper BCK-algebra Lom = {0, a, b} with the hyper operation “cc” which is given in Table 1.

Table 1

Tabular representation of the binary operation cc

cc	0	a	b
0	{0}	{0}	{0}
a	{a}	{0, a}	{0, a}
b	{b}	{a, b}	{0, a, b}

Given a set A = {x, y} of parameters, we define a fuzzy soft set A by Table 2.

Table 2

Tabular representation of A

kap	0	a	b
x	0.9	0.5	0.3
y	0.8	0.4	0.6

It is clear that kap [y] (0) ≥ kap [y] (z) for all z ∈ Lom. But a ⪡ b and kap [y] (a) =0.4 < 0.6 = kap [y] (b).

Definition 3.2. Let A be a fuzzy soft set over Lom. Given a parameter u ∈ A, we say that A is a fuzzy soft BCK-ideal based on u over Lom (briefly, u-fuzzy soft BCK-ideal of Lom) if the fuzzy value set $: \to [0, 1]$ of u satisfies (3.1) and

$\tilde{λ} {[u]}_{*} (x \circ z) \geq \min {\tilde{λ} {[u]}_{*} (x \circ y) \circ z}, \tilde{λ} {[u]}_{*} (y \circ z) .$ (3.2) for all x, y, z ∈ Lom. If A is a u-fuzzy soft BCK-ideal of Lom for all u ∈ A, we say that A is a fuzzy soft BCK-ideal of Lom.

Example 3.3. Consider a BCK-algebra Lom = {0, a, b} with the hyper operation “cc" in Table 3.

Table 3

Cayley table for the binary operation “cc”

cc	0	a	b
0	{0}	{0}	{0}
a	{a}	{0}	{0}
b	{b}	{a, b}	{0, a, b}

Given a set A = {x, y, z} of parameters, we define a fuzzy soft set A by Table 4.

Table 4

Tabular representation of A

kap	0	a	b
x	0.9	0.5	0.3
y	0.8	0.4	0.6
z	0.7	0.7	0.4

Then kap [x] and kap [z] satisfies conditions (3.1) and (3.2). Hence A is a fuzzy soft positive implicative hyper BCK-ideal of Lom based on x and z. But kap [y] does not satisfy the condition (3.1) since a ⪡ b and kap [y] (a) < kap [y] (b), and does not satisfy the condition (3.2) because of $\begin{matrix} \begin{matrix} [x]_{*} (a0) < min {[x]_{*} ((a b) 0), \\ [x]_{*} (b 0)} . \end{matrix} \end{matrix}$ Thus kap [y] are not fuzzy soft positive implicative hyper BCK-ideals based on y over Lom.

Example 3.4. Consider a BCK-algebra Lom = {0, a, b} with the hyper operation “cc" in Table 5.

Table 5

Cayley table for the binary operation “cc”

cc	0	a	b
0	{0}	{0}	{0}
a	{a}	{0, a}	{0, a}
b	{b}	{a, b}	{0, a, b}

Given a set A = {x, y, z} of parameters, we define a fuzzy soft set A by Table 6.

Table 6

Tabular representation of A

kap	0	a	b
x	0.8	0.7	0.6
y	0.5	0.3	0.2
z	0.9	0.6	0.1

Then A is a fuzzy soft positive implicative hyper BCK-ideal of Lom.

Theorem 3.5. Every fuzzy soft positive implicative hyper BCK-ideal is a fuzzy soft BCK-ideal.

Proof. Let A be a fuzzy soft positive implicative hyper BCK-ideal of Lom and let u be any parameter in A. Taking z = 0 in (3.2) and using (2.8) imply that $\begin{array}{l} \tilde{λ} {[u]}_{*} (x \circ z) \geq \min {\tilde{λ} {[u]}_{*} (x \circ y) \circ z}, \tilde{λ} {[u]}_{*} (y \circ z) . \\ \begin{matrix} \tilde{λ} [u] (x) & = & \tilde{λ} {[u]}_{*} (x \circ 0) \\ \geq & \min \tilde{λ} {[u]}_{*} ((x \circ y) \circ 0), \tilde{λ} {[u]}_{*} (y \circ 0)} \\ = & \min \tilde{λ} {[u]}_{*} (x \circ y), \tilde{λ} [u] (y)} . \end{matrix} \end{array}$ Therefore A is a fuzzy soft BCK-ideal of Lom.□

The converse of Theorem 3.5 is not true as seen in the following example.

Example 3.6. Consider a BCK-algebra Lom = {0, a, b} with the hyper operation “cc" in Table 7.

Table 7

Cayley table for the binary operation “cc”

cc	0	a	b
0	{0}	{0}	{0}
a	{a}	{0}	{0}
b	{b}	{a}	{0, a}

Given a set A = {x, y} of parameters, we define a fuzzy soft set A by Table 8.

Table 8

Tabular representation of A

kap	0	a	b
x	0.9	0.5	0.3
y	0.5	0.4	0.4

Then A is a fuzzy soft BCK-ideal based on y over Lom. But it is not a fuzzy soft positive implicative hyper BCK-ideal based on y over Lom since $[y]_{*} (bb) = [y] (a) = 0.4$ and $\begin{matrix} min {[y]_{*} ((b a) b), [y]_{*} (a b)} \\ = [y] (0) = 0.5 . \end{matrix}$

Therefore any fuzzy soft BCK-ideal may not be a fuzzy soft positive implicative hyper BCK-ideal.

Given a fuzzy soft set A over Lom and t ∈ [0, 1], we consider the following set. $3.3 U (; t) : = {x \in∣ (x) \geq t}$ (3.3) where u is a parameter in A, which is called level set of A.

Lemma 3.7. ([16]). Every positive implicative hyper BCK-ideal is a BCK-ideal.

Lemma 3.8. ([4]). A fuzzy soft set A over Lom is a fuzzy soft BCK-ideal of Lom if and only if the set U (kapu ; t) in (3.3) is a BCK-ideal of Lom for all t ∈ [0, 1] and any parameter u in A with U (kapu ; t)≠ ∅.

Theorem 3.9. A fuzzy soft set A over Lom is a fuzzy soft positive implicative hyper BCK-ideal of Lom if and only if the set U (kapu ; t) in (3.3) is a hyper BCK-ideal of Lom for all t ∈ [0, 1] and any parameter u in A with U (kapu ; t)≠ ∅.

Proof. Assume that A is a fuzzy soft positive implicative hyper BCK-ideal of Lom. Let u be a parameter in A and t ∈ [0, 1] be such that U (kapu ; t)≠ ∅. Since 0 ⪡ x for all x ∈ Lom, it follows from (3.1) that kapu (0) ≥ kapu (x) for all x ∈ Lom. Hence $(0) \geq (x)$ for all x ∈ U (kapu ; t), and so kapu (0) ≥ t. Thus 0 ∈ U (kapu ; t). Let x, y, z ∈ Lom be such that $(xy)z ⪡ U (; t)$ and yccz ⊆ U (kapu ; t). For every a ∈ (xccy) ccz, there exists b ∈ U (kapu ; t) such that a ⪡ b. Hence kapu (a) ≥ kapu (b) by (3.1), and thus kapu (a) ≥ t for all a ∈ (xccy) ccz. Let c ∈ xccz. Using (3.2), we have $\begin{matrix} \tilde{λ} [u] (c) & \geq & \tilde{λ} {[u]}_{*} (x \circ z) \\ \geq & \min \tilde{λ} {[u]}_{*} ((x \circ y) \circ z), \tilde{λ} {[u]}_{*} (y \circ z)} \\ \geq & t . \end{matrix}$

Thus c ∈ U (kapu ; t), and so xccz ⊆ U (kapu ; t). Therefore U (kapu ; t) is a positive implicative hyper BCK-ideal of Lom for all t ∈ [0, 1] and any parameter u in A with U (kapu ; t)≠ ∅.

Conversely, assume that U (kapu ; t)≠ ∅ for t ∈ [0, 1] and any parameter u in A. Suppose that U (kapu ; t) is a positive implicative hyper BCK-ideal of Lom. Then U (kapu ; t) is a BCK-ideal of Lom by Lemma 3.7. It follows from Lemma 3.8 that kapu is a BCK-ideal of Lom. Thus the condition (3.1) is valid. Let t = min{ _kapu * ((xccy) ccz) , _kapu * (yccz) }. Then $\tilde{λ} [u] (b) \geq \tilde{λ} {[u]}_{*} ((x \circ y) \circ z) \geq t$ and $\tilde{λ} [u] (c) \geq \tilde{λ} {[u]}_{*} (y \circ z) \geq t$ for all b ∈ (xccy) ccz and c ∈ yccz. Hence $b, c \in U (; t),$ and therefore (xccy) ccz ⊆ U (kapu ; t) and $y z \subseteq U (; t) .$ Using (2.4) and (2.14), we have xccz ⊆ U (kapu ; t), and so $\begin{array}{l} \tilde{λ} {[u]}_{*} (x \circ z) \geq t \\ = \min \tilde{λ} {[u]}_{*} ((x \circ y) \circ z), \tilde{λ} {[u]}_{*} (y \circ z)} . \end{array}$ Consequently, kapu is a fuzzy soft positive implicative hyper BCK-ideal of Lom.□

Corollary 3.10. If a fuzzy soft set A over Lom is a fuzzy soft positive implicative hyper BCK-ideal of Lom, then undersetu ∈ A ⋂ U (kapu ; t) is a positive implicative hyper BCK-ideal of Lom for t ∈ [0, 1].

Lemma 3.11. ([16]). Every BCK-algebra Lom satisfies the following condition.

$3.4 ((x z) (y z)) a ⪡ (x y) a$ (26) for all x, y, z, a ∈ Lom.

Lemma 3.12. ([16]). Let A be a BCK-ideal of Lom. Then IccJ ⊆ A and J ⊆ A imply that I ⊆ A for every non-empty subsets I and J of Lom.

Lemma 3.13. ([14]). Let I be a subset of Lom. If J is a BCK-ideal of Lom such that I ⪡ J, then I is contained in J.

Lemma 3.14. ([16]). For a subset I of Lom such that xccx ⊆ I for all x ∈ Lom, the following assertions are equivalent.

I is a positive implicative hyper BCK-ideal of Lom.

I is a BCK-ideal of Lom such that $\begin{matrix} (x y) z \subseteq I \Rightarrow (x z) (y z) \subseteq I \end{matrix}$ for all x, y, z ∈ Lom.

Question 3.15. Let A be a fuzzy soft positive implicative hyper BCK-ideal of Lom. For any parameter u in A, if U (kapu ; t) is reflexive for all t ∈ Im (kapu), then is the following inequality valid? $\tilde{λ} {[u]}_{*} (x \circ y) \geq \tilde{λ} {[u]}_{*} ((x \circ y) \circ y)$ (3.5) for all x, y ∈ Lom.

The answer to the question above is negative. For example, note that A is a fuzzy soft positive implicative hyper BCK-ideal of Lom based on x and z in Example 3.3. Also U (kap [x] ; t) and U (kap [z] ; t) are reflexive for all t ∈ Im (kapu). But $\begin{matrix} [x]_{*} (b 0) = 0.3 < 0.9 = [x]_{*} ((x 0) 0) . \end{matrix}$ Hence (3.5) is not true.

Proposition 3.16. Let A be a fuzzy soft BCK-ideal of Lom. For any parameter u in A, if A satisfies the condition (3.5), then it satisfies the following condition.

\tilde{λ} {[u]}_{*} ((x \circ z) \circ (y \circ z)) \geq \tilde{λ} {[u]}_{*} ((x \circ y) \circ y)

(3.6)

for all x, y, z ∈ Lom. Moreover if the non-empty level set U (kapu ; t) of A is reflexive for all t ∈ [0, 1], then (3.7) for all x, y, z ∈ Lom.

Proof. Let A be a fuzzy soft BCK-ideal of Lom which satisfies the condition (3.5) for any parameter u in A. Let t = _kapu * ((xccy) ccz). Then $(x y) z \subseteq U (; t) .$ Using (2.2) and Lemma 3.11 induces (3.8) and so ((xcc (yccz)) ccz) ccz ⪡ U (kapu ; t). It follows from Lemma 3.13 that $((x (y z)) z) z \subseteq U (; t)$ and so that (qccz) ccz ⊆ U (kapu ; t) for all q ∈ xcc (yccz). Using the condition (3.5), we get $\tilde{λ} {[u]}_{*} (q \circ z) \geq \tilde{λ} {[u]}_{*} ((q \circ z) \circ z) \geq t,$ and so qccz ⊆ U (kapu ; t) for all q ∈ xcc (yccz). Hence $\begin{matrix} (x z) (y z) & = (x (y z)) z \\ = q \in x (y z) ⋃ q z \\ \subseteq U (; t), \end{matrix}$ and therefore $\tilde{λ} {[u]}_{*} ((x \circ z) \circ (y \circ z)) \geq t = \tilde{λ} {[u]}_{*} ((x \circ y) \circ z) .$ This proves (3.6). Suppose that U (kapu ; t) of A is reflexive for all t ∈ [0, 1]. For any x, y, z ∈ Lom, put $s = \min {\tilde{λ} [u] (z), \tilde{λ} {[u]}_{*} (((x \circ y) \circ y) \circ z)} .$ Then z ∈ U (kapu ; s) and $((x z) y) y = ((x y) y) z \subseteq U (; s) .$ Thus (qccy) ccy ⊆ U (kapu ; s), which implies from Lemma 3.14 that $(q y) (y y) \subseteq U (; s)$ for all q ∈ xccz. Thus $((x z) y) (y y) \subseteq U (; s),$ and so (xccy) ccz = (xccz) ccy ⊆ U (kapu ; s) by Lemma 3.12 and (H2). Since z ∈ U (kapu ; s), we have xccy ⊆ U (kapu ; s) by Lemma 3.12. Hence $\begin{array}{l} \tilde{λ} {[u]}_{*} (x \circ y) \geq s \\ = \min {\tilde{λ} [u] (z), \tilde{λ} {[u]}_{*} (((x \circ y) \circ y) \circ z)} \end{array}$ for all x, y, z ∈ Lom. This completes the proof.□

Using the notion of positive implicative hyper BCK-ideal of Lom, we establish a fuzzy soft weak BCK-ideal.

Theorem 3.17. Let I be a positive implicative hyper BCK-ideal of Lom and let z ∈ Lom. For a fuzzy soft set A over Lom and any parameter u in A, if we define the fuzzy value set kapu by

: \to [0, 1], x \mapsto {\begin{matrix} t & if x \in I_{z}, \\ s & otherwise, \end{matrix}

(3.9)

where t > s in [0, 1] and I_z : = {y ∈ Lom ∣ yccz ⊆ I}, then A is a u-fuzzy soft weak BCK-ideal of Lom.

Proof. It is clear that kapu (0) ≥ kapu (x) for all x ∈ Lom. Let x, y ∈ Lom. If y ∉ I_z, then kapu (y) = s and so $\tilde{λ} [u] (x) \geq s = \min {\tilde{λ} [u] (y), \tilde{λ} {[u]}_{*} (x \circ y)} .$ (3.10) If xccy ⊈ I_z, then there exists a ∈ xccy \ I_z, and thus kapu (a) = s. Hence $\min {\tilde{λ} [u] (y), \tilde{λ} {[u]}_{*} (x \circ y)} = s \leq \tilde{λ} [u] (x) .$ (3.11) Assume that xccy ⊆ I_z and y ∈ I_z. Then (xccy) ccz ⊆ I and yccz ⊆ I, which imply that (xccy) ccz ⪡ I and yccz ⊆ I. It follows from (2.14) that xccz ⊆ I, i.e., x ∈ I_z. Thus $\tilde{λ} [u] (x) = t \geq \min {\tilde{λ} [u] (y), \tilde{λ} {[u]}_{*} (x \circ y)},$ and hence kapu is a fuzzy soft weak BCK-ideal of Lom. Therefore A is a u-fuzzy soft weak BCK-ideal of Lom.□

Lemma 3.18. ([17]). Every reflexive BCK-ideal I of Lom satisfies the following implication.

(x y) \cap I \neq \emptyset \Rightarrow x y \subseteq I

(3.12)

for all x, y ∈ Lom.

Theorem 3.19. If A is a fuzzy soft positive implicative hyper BCK-ideal of Lom in which the non-empty level set U (kapu ; t) of A is reflexive for all t ∈ Im (kapu), then the set

(3.13)

is a BCK-ideal of Lom for all z ∈ Lom.

Proof. Obviously 0 ∈ _kapuz. Let x, y ∈ Lom be such that xccy ⊆ _kapuz and y ∈ _kapuz. Then $(x y) z \subseteq U (; t)$ and yccz ⊆ U (kapu ; t) for all t ∈ Im (kapu). Using (2.4), we know that (xccy) ccz ⪡ U (kapu ; t). Since U (kapu ; t) is a positive implicative hyper BCK-ideal of Lom, it follows from (2.14) that xccz ⊆ U (kapu ; t), that is, x ∈ _kapuz. This shows that _kapuz is a weak BCK-ideal of Lom. Let x, y ∈ Lom be such that xccy ⪡ _kapuz and y ∈ _kapuz, and let a ∈ xccy. Then there exists b ∈ _kapuz such that a ⪡ b, that is, 0 ∈ accb. Thus (accb)∩ U (kapu ; t) ≠ ∅. Since U (kapu ; t) is a reflexive BCK-ideal of Lom, it follows from (H1) and Lemma 3.18 that $(a z) (b z) ⪡ a b \subseteq U (; t)$ and so that accz ⊆ U (kapu ; t) since bccz ⊆ U (kapu ; t). Hence a ∈ _kapuz, and so xccy ⊆ _kapuz. Since _kapuz is a weak BCK-ideal of Lom, we get x ∈ _kapuz. Consequently _kapuz is a BCK-ideal of Lom.□

The following example shows that any positive implicative hyper BCK-ideal is neither S-reflexive nor a strong BCK-ideal.

Example 3.20. Consider the BCK-algebra Lom = {0, a, b} in Example 3.3. Then the set I : = {0, a} is a positive implicative hyper BCK-ideal of Lom. But it is not S-reflexive since (bcca)∩ I ≠ ∅ but bcca ⊈ I. Also, I is not a strong BCK-ideal of Lom since (bcca)∩ I ≠ ∅ and a ∈ I, but b ∉ I.

Using the notion of positive implicative hyper BCK-ideal of Lom, we establish a fuzzy soft strong BCK-ideal.

Lemma 3.21. Every S-reflexive positive implicative hyper BCK-ideal is a strong BCK-ideal.

Proof. Let I be an S-reflexive positive implicative hyper BCK-ideal of Lom and let x, y ∈ Lom be such that (xccy)∩ I ≠ ∅ and y ∈ I. Then (xccy) cc0 = xccy ⊆ I since I is S-reflexive and Acc0 = A for any subset A of Lom. It follows from (2.14) that (xccy) cc0 ⪡ I. Since ycc0 ⊆ I, we have {x} = xcc0 ⊆ I and so x ∈ I. Therefore I is a strong BCK-ideal of Lom.□

The following example shows that the converse of Lemma 3.21 is not true in general.

Example 3.22. Consider a BCK-algebra Lom = {0, a, b, c} with the hyper operation “cc" in Table 9.

Table 9

Cayley table for the binary operation “cc”

cc	0	a	b	c
0	{0}	{0}	{0}	{0}
a	{a}	{0}	{0}	{a}
b	{b}	{a}	{0}	{b}
c	{c}	{c}	{c}	{0}

Then I : = {0, c} is a strong BCK-ideal and S-reflexive. But it is not a positive implicative hyper BCK-ideal since (bcca) cca ⪡ I and acca ⊆ I but bcca ⊈ I.

Lemma 3.23. ([4]). Let A be a fuzzy soft set over Lom such that

(\forall T \subseteq H) (\exists x_{0} \in T) (\tilde{λ} [u] (x_{0}) = \tilde{λ} {[u]}^{*} (T))

(3.14)

where u is any parameter in A. If the set U (kapu ; t) in (3.3) is a strong BCK-ideal of Lom for all t ∈ [0, 1] with U (kapu ; t)≠ ∅, then A is a fuzzy soft strong BCK-ideal of Lom.

Using Lemmas 3.21 and 3.23, we have the following theorem.

Theorem 3.24. Let A be a fuzzy soft set over Lom satisfying the condition (3.14). If the set U (kapu ; t) in (3.3) is an S-reflexive positive implicative hyper BCK-ideal of Lom for all t ∈ [0, 1] with U (kapu ; t)≠ ∅, then A is a fuzzy soft strong BCK-ideal of Lom.

The following example shows that the converse of Theorem 3.24 is not true in general.

Example 3.25. Consider a BCK-algebra Lom = {0, a, b} with the hyper operation “cc" in Table 10.

Table 10

Cayley table for the binary operation “cc”

cc	0	a	b
0	{0}	{0}	{0}
a	{a}	{0}	{a}
b	{b}	{b}	{0, b}

Given a set A = {x, y} of parameters, we define a fuzzy soft set A by Table 11.

Table 11

Tabular representation of A

kap	0	a	b
x	0.9	0.5	0.3
y	0.8	0.6	0.6

It is routine to verify that A is a fuzzy soft strong BCK-ideal of Lom. If t > 0.6, then the set U (kap [x] ; t) = {0} is not a positive implicative hyper BCK-ideal of Lom since (0ccb) ccb = {0} ⪡ U (kap [x] ; t), bccb = {0, b} ⊈ U (kap [x] ; t) and $0 b = {0} \subseteq U ([x]; t) .$

Conclusions

We have applied the notion of fuzzy soft sets by Maji et al. to the theory of hyper BCK-algebras. We have introduced the notion of fuzzy soft positive implicative hyper BCK-ideal, and have investigated several properties. We have discussed the relation between fuzzy soft positive implicative hyper BCK-ideal and fuzzy soft hyper BCK-ideal, and have provided characterizations of fuzzy soft positive implicative hyper BCK-ideal. We have established a fuzzy soft weak (strong) hyper BCK-ideal by using the notion of positive implicative hyper BCK-ideal. We hope in the forthcoming researches and papers we continue this idea and define some new notions.

Footnotes

Acknowledgments

The authors wish to thank the anonymous reviewers for their valuable suggestions.

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