Some types of pairwise soft sets and the associated soft topologies

Abstract

In this paper, we introduce the notions of pairwise soft sub kernel, and pairwise ∨-soft sets in a soft bitopological space (X, η₁, η₂, E). Also, we study the fundamental properties of pairwise ∨-soft sets and we investigate the associated soft topology η_p∨. Moreover, we introduce the notions of generalized pairwise Λ(∨)-soft sets and we investigate their basic properties. Also, we define a soft closure operator on the family of all generalized pairwise Λ-soft sets and generate, in usual manner, an Alexandroff soft topology η_gpΛ on X which is finer than η_p∨. Furthermore, we prove that (X, η_gpΛ, E) is always a soft $T_{\frac{1}{2}}$ space. In addition, we introduce characterizations of pairwise soft $T_{\frac{1}{2}}^{*}$ by using generalized pairwise Λ-soft sets and we show that the concepts of generalized pairwise Λ-soft set and generalized pairwise closed soft set are independent.

Keywords

Soft sets soft topology soft bitopology soft bitopological spaces pairwise soft sets

1 Introduction

In 1999, Molodtsov [15] introduced the concept of soft sets as a mathematical tool for dealing with uncertainties. Shabir and Naz [20] introduced the notion of soft topological spaces which are defined over an initial universe with a fixed set of parameters. They also studied some of basic properties of soft topological spaces. After that, many researches (see, for example, [2 , 16– 18]) introduced and studied new notions of soft topological spaces. Ittanagi [3] introduced the notion of soft bitopological space which is defined over an initial universal set X with fixed set of parameters E, also he introduced some types of soft separation axioms in soft bitopological space. Kandil et al. [11] introduced some structures of soft bitopological space. They defined some basic notions in soft bitopological space as pairwise open (closed) soft set, pairwise soft closure (resp. interior, kernel) operator and defined some soft sets as pΛ-soft set, and pλ-closed soft set and established some of their properties. They show that the family of all pairwise open soft sets is a supra soft topology $η_{12} (= {(G_{1}, E) \tilde{\cup} (G_{2}, E) : (G_{i}, E) \in η_{i}, i = 1, 2})$ which is containing η₁, η₂ but it is not soft topology in general. Moreover, they proved that the family of all pΛ-soft sets defines an Alexandroff soft topology η_pΛ which is finer than η₁, η₂ and containing η₁₂.

Recently, Kandil et al. [10] introduced the concept of generalized pairwise closed soft sets and the associated pairwise soft separation axioms namely, ${PST}_{\frac{1}{2}}^{*}$ and ${PSR}_{0}^{*}$ .

This paper a continuation of [10, 11]. We introduce the notion of pairwise soft sub kernel, and pairwise ∨-soft sets in a soft bitopological spaces and we study the fundamental properties of pairwise ∨-soft sets and we investigated the soft topology defined by this family of soft sets. Moreover, we introduce the notions of generalized pairwise Λ-soft sets and generalized pairwise ∨-soft sets. Also, we investigate their basic properties. Moreover, we define a soft closure operator on that the family of all generalized pairwise Λ-soft sets and generate, in usual manner, an Alexandroff soft topology η_gpΛ which is finer than η_p∨. Furthermore, we prove that (X, η_pgΛ, E) is always a soft $T_{\frac{1}{2}}$ space. In addition, we introduce the characterization of pairwise soft $T_{\frac{1}{2}}^{*}$ by using generalized pairwise Λ-soft sets.

2 Preliminaries

In this section, we collect some needed definitions and theorems of the material used in this paper.

Definition 2.1. [17] Let X be an initial universe and E be a set of parameters. Let P (X) denote the power set of X. A pair (F, E) is called a soft set over X, where F is a mapping given by F : E → P (X). A soft set can also be defined by the set of ordered pairs $(F, E) = {(e, F (e)) : e \in E, F : E \to P (X)} .$

From now on, SS (X) _E denotes the family of all soft sets over X with a fixed set of parameters E.

Definition 2.2. [13] Let (F, E) , (G, E) ∈ SS (X) _E. Then,

(F, E) is called a soft subset of (G, E), denoted by $(F, E) \tilde{\subseteq} (G, E)$ , if F (e) ⊆ G (e), ∀e ∈ E. In this case, (F, E) is said to be a soft subset of (G, E) and (G, E) is said to be a soft superset of (F, E).

Two soft sets (F, E) and (G, E) over a common universe set X are said to be equal, denoted by (F, E) = (G, E), if F (e) = G (e), ∀ e ∈ E.

The union of two soft sets (F, E) and (G, E) over the common universe X, denoted by $(F, E) \tilde{\cup} (G, E)$ , is the soft set (H, E), where H (e) = F (e) ∪ G (e), ∀e ∈ E.

The intersection of two soft sets (F, E) and (G, E) over the common universe X, denoted by $(F, E) \tilde{\cap} (G, E)$ , is the soft set (M, E), where M (e) = F (e) ∩ G (e), ∀e ∈ E.

A soft set (F, E) over X is said to be a NULL soft set, denoted by $(\tilde{φ}, E)$ or $\tilde{φ}$ , if F (e) = φ, ∀e ∈ E.

A soft set (F, E) over X is said to be an absolute soft set, denoted by $(\tilde{X}, E)$ or $\tilde{X}$ , if F (e) = X, ∀e ∈ E.

Definition 2.3. [1] The complement of a soft set (F, E), denoted by (F, E) ^c, is defined by (F, E) ^c = (F^c, E), where F^c : E → P (X) is a mapping given by F^c (e) = X \ F (e), ∀ e ∈ E.

Clearly, (F^c) ^c is the same as F, ((F, E) ^c) ^c = (F, E), $(\tilde{φ}, E)^{c} = (\tilde{X}, E)$ and $(\tilde{X}, E)^{c} = (\tilde{φ}, E)$ .

Definition 2.4. [20] The difference of two soft sets (F, E) and (G, E) over the common universe X, denoted by (F, E) \ (G, E), is the soft set (H, E), where for all e ∈ E, H (e) = F (e) \ G (e). Clearly, $(F, E) \ (G, E) = (F, E) \tilde{\cap} (G, E)^{c}$ .

For more details about the properties of the union, the intersection and the complement of soft sets can you see in [6 , 23].

Definition 2.5. [16, 21] The soft set (F, E) ∈ SS (X) _E is called a soft point in $(\tilde{X}, E)$ if there exist x ∈ X and e ∈ E such that F (e) = {x} and F (e′) = φ for each e′ ∈ E \ {e}. This soft point is denoted by (x_e, E) or x_e, i.e., x_e : E → P (X) is a mapping defined by $x_{e} (a) = {\begin{matrix} {x} if e = a, \\ φ if e \neq a \end{matrix} for all a \in E .$

The set of all soft points in $(\tilde{X}, E)$ is denoted by ξ (X) _E.

Definition 2.6. [21] The soft point (x_e, E) is said to be belonging to the soft set (G, E), denoted by $x_{e} \tilde{\in} (G, E)$ , if x_e (e) ⊆ G (e), i.e., {x} ⊆ G (e). Clearly, $x_{e} \tilde{\in} (G, E)$ if and only if $(x_{e}, E) \tilde{\subseteq} (G, E)$ .

Definition 2.7. [21] The two soft points x_{e
₁}, y_{e
₂} over X are said to be equal if x = y and e₁ = e₂. Thus, x_{e
₁} ≠ y_{e
₂} iff x ≠ y or e₁ ≠ e₂.

Proposition 2.8. [21] The union of any collection of soft points can be considered as a soft set and every soft set can be expressed as a union of all soft points belonging to it, i.e., $(G, E) = ⋃^{˜} {x_{e} : x_{e} \tilde{\in} (G, E)}$ .

Proposition 2.9. [21] Let (G, E), (H, E) be two soft sets over X. Then,

$x_{e} \tilde{\in} (G, E) \Leftrightarrow x_{e} \tilde{\notin} (G, E)^{c}$ .

$x_{e} \tilde{\in} (G, E) \tilde{\cup} (H, E) \Leftrightarrow x_{e} \tilde{\in} (G, E)$ or $x_{e} \tilde{\in} (H, E)$ .

$x_{e} \tilde{\in} (G, E) \tilde{\cap} (H, E) \Leftrightarrow x_{e} \tilde{\in} (G, E)$ and $x_{e} \tilde{\in} (H, E)$ .

$(G, E) \tilde{\subseteq} (H, E) \Leftrightarrow [x_{e} \tilde{\in} (G, E) \Rightarrow x_{e} \tilde{\in} (H, E)]$ .

Definition 2.10. [20] Let η be a collection of soft sets over a universe X with a fixed set of parameters E, i.e., η ⊆ SS (X) _E. Then η is called a soft topology on X if it satisfies the following axioms:

$(\tilde{X}, E), (\tilde{φ}, E) \in η$ ,

The union of any number of soft sets in η belongs to η,

The intersection of any two soft sets in η belongs to η.

The triple (X, η, E) is called a soft topological space. Any member of η is said to be an open soft set in (X, η, E). A soft set (F, E) over X is said to be a closed soft set in (X, η, E), if its complement (F, E) ^c is an open soft set in (X, η, E).

We denote the family of all closed soft sets by η^c.

Definition 2.11. [20] Let (X, η, E) be a soft topological space and (F, E) ∈ SS (X) _E. The soft closure of (F, E), denoted by scl_η (F, E), is the intersection of all closed soft super sets of (F, E), i.e, ${scl}_{η} (F, E) = ⋂^{˜} {(H, E) \in {SC}_{η} (X) : (F, E) \tilde{\subseteq} (H, E)})$ . Clearly scl_η (F, E) is the smallest closed soft set over X which contains (F, E).

For more details about the properties of soft topological spaces can you see in [2 , 23].

Theorem 2.12. [22] Let (X, η, E) be a soft topological space. Then, $x_{e} \tilde{\in} {scl}_{η} (G, E) \Leftrightarrow (O, E) \tilde{\cap} (G, E) \neq (\tilde{φ}, E),$ ∀ (O, E) ∈ η (x_e), where η (x_e) denote the family of all open soft sets which contains x_e.

Definition 2.13. [12] A soft set (G, E) in a soft topological space (X, η, E) is called a generalized closed soft set [briefly, g-closed soft set] if ${scl}_{η} (G, E) \tilde{\subseteq} (H, E)$ whenever $(G, E) \tilde{\subseteq} (H, E)$ and (H, E) ∈ η.

Definition 2.14. [12] A a soft topological space (X, η, E) is called a soft $T_{\frac{1}{2}}$ [briefly, ${ST}_{\frac{1}{2}}$ ] if every g-closed soft set is a closed soft set.

Definition 2.15. [5] Let μ be a collection of soft sets over X [i.e., μ ⊆ SS (X) _E]. Then, μ is called a supra soft topology on X if it satisfies the following axioms:

$(\tilde{X}, E)$ , $(\tilde{φ}, E) \in μ$ ,

The union of any number of soft sets in μ belongs to μ.

Definition 2.16. [3] A quadrable system (X, η₁, η₂, E) is called a soft bitopological space [briefly, sbts], where η₁, η₂ are arbitrary soft topologies on X with a fixed set of parameters E.

Definition 2.17. [11] Let (X, η₁, η₂, E) be a sbts. A soft set (G, E) over X is said to be a pairwise open soft set in (X, η₁, η₂, E) [briefly, p-open soft set] if there exist an open soft set (G₁, E) in η₁ and an open soft set (G₂, E) in η₂ such that $(G, E) = (G_{1}, E) \tilde{\cup} (G_{2}, E)$ . A soft set (G, E) over X is said to be a pairwise closed soft set in (X, η₁, η₂, E) [briefly, p-closed soft set] if its complement is a p-open soft set in (X, η₁, η₂, E). Clearly, a soft set (F, E) over X is a p-closed soft set in (X, η₁, η₂, E) if there exist an η₁-closed soft set (F₁, E) and an η₂-closed soft set (F₂, E) such that $(F, E) = (F_{1}, E) \tilde{\cap} (F_{2}, E)$ .

The family of all p-open (p-closed) soft sets in sbts (X, η₁, η₂, E) is denoted by η₁₂ ( $η_{12}^{c}$ ), respectively.

Theorem 2.18. [11] Let (X, η₁, η₂, E) be a sbts. The family η₁₂ of all p-open soft sets η₁₂ is a supra soft topology on X, where

$η_{12} = {(G, E) = (G_{1}, E) \tilde{\cup} (G_{2}, E) : (G_{i}, E) \in η_{i}, i = 1, 2}$ . The triple (X, η₁₂, E) is the supra soft topological space associated to the soft bitopological space (X, η₁, η₂, E).

Definition 2.19. [11] Let (X, η₁, η₂, E) be a sbts and let (G, E) ∈ SS (X) _E. The pairwise soft closure of (G, E), denoted by scl_p (G, E), is the intersection of all p-closed soft supersets of (G, E), i.e., ${scl}_{p} (G, E) = ⋂^{˜} {(F, E) \in η_{12}^{c} : (G, E) \tilde{\subseteq} (F, E)} .$

Clearly, scl_p (G, E) is the smallest p-closed soft set contains (G, E). For more details about the properties of pairwise soft closure operator see in [11].

Definition 2.20. [11] Let (X, η₁, η₂, E) be a sbts and let (G, E) ∈ SS (X) _E. The pairwise soft interior of (G, E), denoted by sint_p (G, E), is the union of all p-open soft subsets of (G, E), i.e., ${sint}_{p} (G, E) = ⋃^{˜} {(H, E) \in η_{12} : (H, E) \tilde{\subseteq} (G, E)} .$

Clearly, sint_p (G, E) is the largest p-open soft set contained in (G, E). For more details about the properties of pairwise soft interior operator see in [11].

Definition 2.21. [11] Let (X, η₁, η₂, E) be a sbts and let (G, E) ∈ SS (X) _E. The pairwise soft kernel of (G, E) [briefly, sker_p (G, E)], is the intersection of all p-open soft supersets of (G, E), i.e., ${sker}_{p} (G, E) = ⋂^{˜} {(H, E) \in η_{12} : (G, E) \tilde{\subseteq} (H, E)} .$

Theorem 2.22. [11] Let (X, η₁, η₂, E) be a sbts and let (G, E) , (H, E) ∈ SS (X) _E. Then,

${sker}_{p} (\tilde{X}, E) = (\tilde{X}, E)$ and ${sker}_{p} (\tilde{φ}, E) = (\tilde{φ}, E)$ .

$(G, E) \tilde{\subseteq} {sker}_{p} (G, E)$ .

$(G, E) \tilde{\subseteq} (H, E) \Rightarrow {sker}_{p} (G, E) \tilde{\subseteq} {sker}_{p} (H, E)$ .

(G, E) ∈ η₁₂ ⇒ sker_p (G, E) = (G, E).

sker_p [sker_p (G, E)] = sker_p (G, E).

${sker}_{p} [⋂^{˜} {(H_{i}, E) : i \in Δ}] \tilde{\subseteq} ⋂^{˜} {{sker}_{p} (H_{i}, E) : i \in Δ}$ .

${sker}_{p} [⋃^{˜} {(G_{i}, E) : i \in Δ}] = ⋃^{˜} {{sker}_{p} (G_{i}, E) : i \in Δ}$ .

Definition 2.23. [11] A soft set (G, E) is said to be a pairwise Λ- soft set in a soft bitopological space (X, η₁, η₂, E) [briefly, pΛ-soft set] if sker_p (G, E) = (G, E).

Theorem 2.24. [11] Every p-open soft set is a pΛ-soft set.

Theorem 2.25. [11] Let (X, η₁, η₂, E) be a sbts. The class of all pΛ-soft sets is an Alexandroff soft topology on X. This soft topology, we denoted by η_PΛ. The triple (X, η_pΛ, E) is the soft topological space associated to the soft bitopological space (X, η₁, η₂, E), induced by the family of pΛ-soft sets.

Theorem 2.26. [11] Let (X, η₁, η₂, E) be a sbts. Then, $η_{1} \cup η_{2} \subseteq η_{12} \subseteq η_{p Λ} \subseteq SS (X)_{E} .$

Definition 2.27. [10] Let (X, η₁, η₂, E) be a sbts and let (G, E) ∈ SS (X) _E. A soft set (G, E) is said to be a generalized pairwise closed soft set [briefly, gp-closed soft set] if ${scl}_{p} (G, E) \tilde{\subseteq} (H, E)$ whenever $(G, E) \tilde{\subseteq} (H, E)$ and (H, E) is a p-open soft set.

Theorem 2.28. [10] A soft topological space (X, η, E) is a soft $T_{\frac{1}{2}}$ if and only if every soft point either open soft set or closed soft set.

Definition 2.29. [10] A a soft bitopological space (X, η₁, η₂, E) is called a pairwise soft $T_{\frac{1}{2}}^{*}$ [briefly, ${PST}_{\frac{1}{2}}^{*}$ ] if every gp-closed soft set is a p-closed soft set.

Theorem 2.30. [10] In any sbts (X, η₁, η₂, E), every soft point either p-closed soft set or its complement is a gp-closed soft set.

Theorem 2.31. [10] Let (X, η₁, η₂, E) be a soft bitopological space. Then (X, η₁, η₂, E) is a ${PST}_{\frac{1}{2}}^{*}$ if and only if every soft point in (X, η₁, η₂, E) either p-open or p-closed soft set.

3 Pairwise soft sub kernel operator

Definition 3.1. Let (X, η₁, η₂, E) be a sbts and let (G, E) ∈ SS (X) _E. The pairwise soft sub Kernel of (G, E) [briefly, ${sker}_{p}^{*} (G, E)$ ] is the union of all p-closed soft subsets of (G, E), i.e., ${sker}_{p}^{*} (G, E) = ⋃^{˜} {(F, E) \in η_{12}^{c} : (F, E) \tilde{\subseteq} (G, E)},$ where $η_{12}^{c}$ is the family of all p-closed soft sets.

Remark 3.2. The pairwise soft sub kernel of (G, E) is not p-closed soft set in general as shown in the following example.

Example 3.3. Let X = {x, y, z, w}, E = {e₁, e₂} and let

(G, E) = {(e₁, {x, z, w}) , (e₂, {x, z, w})}. We will find the soft sub kernel of (G, E) with respect to the soft bitoplogical space (X, η₁, η₂, E),

where $\begin{matrix} η_{1} = {(\tilde{φ}, E), (\tilde{X}, E), (G_{1}, E), (G_{2}, E)}, \\ η_{2} = {(\tilde{φ}, E), (\tilde{X}, E), (H_{1}, E), (H_{2}, E)}, \end{matrix}$ such that $\begin{matrix} (G_{1}, E) = {(e_{1}, {x, y}), (e_{2}, {y, z})}, \\ (G_{2}, E) = {(e_{1}, {z, w}), (e_{2}, {x, w})}, \\ (H_{1}, E) = {(e_{1}, {y}), (e_{2}, {y, w})}, \\ (H_{2}, E) = {(e_{1}, {x, y}), (e_{2}, X)} . \end{matrix}$

It is easy to verify that $\begin{matrix} η_{12} & = & {(\tilde{φ}, E), (\tilde{X}, E), (G_{1}, E), (G_{2}, E), (H_{1}, E), \\ (H_{2}, E), (P_{1}, E), (P_{2}, E)}, \end{matrix}$ where $\begin{matrix} (P_{1}, E) = {(e_{1}, {x, y}), (e_{2}, {y, z, w})}, \\ (P_{2}, E) = {(e_{1}, {y, z, w}), (e_{2}, {x, y, w})} . \end{matrix}$

Therefore, the family of all p-closed soft sets is $\begin{matrix} η_{12}^{c} & = & {(\tilde{φ}, E), (\tilde{X}, E), (G_{1}, E)^{c}, (G_{2}, E)^{c}, \\ (H_{1}, E)^{c}, (H_{2}, E)^{c}, (P_{1}, E)^{c}, (P_{2}, E)^{c}}, \end{matrix}$ where $\begin{matrix} (G_{1}, E)^{c} = {(e_{1}, {z, w}), (e_{2}, {x, w})}, \\ (G_{2}, E)^{c} = {(e_{1}, {x, y}), (e_{2}, {y, z})}, \\ (H_{1}, E)^{c} = {(e_{1}, {x, z, w}), (e_{2}, {x, z})}, \\ (H_{2}, E)^{c} = {(e_{1}, {z, w}), (e_{2}, φ)}, \\ (P_{1}, E)^{c} = {(e_{1}, {z, w}), (e_{2}, {x})}, \\ (P_{2}, E)^{c} = {(e_{1}, {x}), (e_{2}, {z})} . \end{matrix}$ It follows that, $\begin{matrix} {sker}_{p}^{*} (G, E) & = & {(e_{1}, {x, z, w}), (e_{2}, {x, z, w})} \\ = & (G, E) . \end{matrix}$

It is clear that ${sker}_{p}^{*} (G, E)$ is not a p-closed soft set.

Theorem 3.4. Let (X, η₁, η₂, E) be a sbts and (G, E) ∈ SS (X) _E. Then,

$x_{e} \tilde{\in} {sker}_{p}^{*} (G, E) \Leftrightarrow \exists (F, E) \in η_{12}^{c}$ such that $x_{e} \tilde{\in} (F, E) \tilde{\subseteq} (G, E)$ .

Proof. Immediate from Definition 3.1.

Theorem 3.5. Let (X, η₁, η₂, E) be a sbts and let (G, E) , (H, E) ∈ SS (X) _E. Then,

${sker}_{p}^{*} (\tilde{X}, E) = (\tilde{X}, E)$ and ${sker}_{p}^{*} (\tilde{φ}, E) = (\tilde{φ}, E)$ .

${sker}_{p}^{*} (G, E) \tilde{\subseteq} (G, E)$ .

$(G, E) \tilde{\subseteq} (H, E) \Rightarrow {sker}_{p}^{*} (G, E) \tilde{\subseteq} {sker}_{p}^{*} (H, E)$ .

$(G, E) \in η_{12}^{c} \Rightarrow {sker}_{p}^{*} (G, E) = (G, E)$ .

${sker}_{p}^{*} [{sker}_{p}^{*} (G, E)] = {sker}_{p}^{*} (G, E)$ .

$⋃^{˜} {{sker}_{p}^{*} (H_{i}, E) : i \in Δ} \tilde{\subseteq} {sker}_{p}^{*} [⋃^{˜} {(H_{i}, E) : i \in Δ}]$ .

${sker}_{p}^{*} [⋂^{˜} {(G_{i}, E) : i \in Δ}] = ⋂^{˜} {{sker}_{p}^{*} (G_{i}, E) : i \in Δ}$ .

Proof. From (1) to (4), it is obvious from Definition 3.1.

(5) Since ${sker}_{p}^{*} (G, E) \tilde{\subseteq} (G, E)$ , then ${sker}_{p}^{*} [{sker}_{p}^{*} (G, E)] \tilde{\subseteq} {sker}_{p}^{*} (G, E)]$ . To prove the inverse inclusion, let $x_{e} \tilde{\in} {sker}_{p}^{*} (G, E)$ . Then, there exists $(F, E) \in η_{12}^{c}$ such that $x_{e} \tilde{\in} (F, E) \tilde{\subseteq} (G, E)$ . Therefore, by using (3) and (4), we have $x_{e} \tilde{\in} (F, E) \tilde{\subseteq} {sker}_{p}^{*} (G, E)$ . Hence, $x_{e} \tilde{\in} ⋃^{˜} {(F, E) \in η_{12}^{c} : (F, E) \tilde{\subseteq} {sker}_{p}^{*} (G, E)}$ , i.e., $x_{e} \tilde{\in} {sker}_{p}^{*} [{sker}_{p}^{*} (G, E)]$ . Therefore, (5) holds.

(6) Since $(H_{i}, E) \tilde{\subseteq} ⋃^{˜} {(H_{i}, E) : i \in Δ} \forall i \in Δ$ ,then ${sker}_{p}^{*} (H_{i}, E) \tilde{\subseteq} {sker}_{p}^{*} [⋃^{˜} {(H_{i}, E) : i \in Δ}] \forall i \in Δ$ .Therefore, $⋃_{i \in Δ} {sker}_{p}^{*} (H_{i}, E) \tilde{\subseteq} {sker}_{p}^{*} [{⋃^{˜}}_{i \in Δ} (H_{i}, E)]$ .

(7) Since ${⋂^{˜}}_{i \in Δ} (G_{i}, E) \subseteq (G_{i}, E)$ , then ${sker}_{p}^{*} [{⋂^{˜}}_{i \in Δ} (G_{i}, E)] \subseteq {sker}_{p}^{*} (G_{i}, E)$ for all i ∈ Δ implies ${sker}_{p}^{*} [{⋂^{˜}}_{i \in Δ} (G_{i}, E)] \subseteq {⋂^{˜}}_{i \in Δ} {sker}_{p}^{*} (G_{i}, E)$ . To prove the inverse inclusion, let $x_{e} \tilde{\in} {⋂^{˜}}_{i \in Δ} {sker}_{p}^{*} (G_{i}, E)$ . Then, $x_{e} \tilde{\in} {sker}_{p}^{*} (G_{i}, E)$ ∀ i ∈ Δ. Thus, by Theorem 3.4, for all index i ∈ Δ there exists $(F_{i}, E) \in η_{12}^{c}$ such that $x_{e} \tilde{\in} (F_{i}, E) \tilde{\subseteq} (G_{i}, E)$ . We take $(F, E) = ⋂^{˜} {(F_{i}, E) : i \in Δ}$ . It follows that $(F, E) \in η_{12}^{c}$ and $x_{e} \tilde{\in} (F, E)$ . Since $(F_{i}, E) \tilde{\subseteq} (G_{i}, E)$ , then ${⋂^{˜}}_{i \in Δ} (F_{i}, E) \tilde{\subseteq} {⋂^{˜}}_{i \in Δ} (G_{i}, E)$ which implies that $x_{e} \tilde{\in} (F, E) \tilde{\subseteq} {⋂^{˜}}_{i \in Δ} (G_{i}, E)$ .

Therefore, $x_{e} \tilde{\in} {sker}_{p}^{*} [{⋂^{˜}}_{i \in Δ} (G_{i}, E)]$ . Hence, (7) holds.

Remark 3.6. ${sker}_{p}^{*} [(H, E) \tilde{\cup} (G, E)] \neq {sker}_{p}^{*} (H, E) \tilde{\cup} {sker}_{p}^{*} (G, E)$ in general as shown in the following example.

Example 3.7. In Example 3.3, let (H, E) = {(e₁, {x, y}) , (e₂, {x, z})} and (G, E) = {(e₁, {z, w}) , (e₂, {x, y, w})}.

Then, ${sker}_{p}^{*} (H, E) = {(e_{1}, {x}), (e_{2}, {z})}$ and ${sker}_{p}^{*} (G, E) = {(e_{1}, {z, w}), (e_{2}, {x, w})}$ . It followsthat, ${sker}_{p}^{*} (H, E) \tilde{\cup} {sker}_{p}^{*} (G, E) = {(e_{1}, {x, z, w}), (e_{2}, {x, z, w})}$ . On the other hand, ${sker}_{p}^{*} [(H, E) \tilde{\cup} (G, E)] = {sker}_{p}^{*} (\tilde{X}, E) = (\tilde{X}, E)$ .

It is clear that ${sker}_{p}^{*} [(H, E) \tilde{\cup} (G, E)] \neq {sker}_{p}^{*} (H, E) \tilde{\cup} {sker}_{p}^{*} (G, E)$ .

4 Pairwise V-soft sets

Definition 4.1. A soft set (G, E) is said to be a pairwise ∨-soft set [briefly, p∨-soft set] in a sbts (X, η₁, η₂, E) if ${sker}_{p}^{*} (G, E) = (G, E)$ . We denote the family of all p∨-soft sets by P ∨ S (X, η₁, η₂) _E or P ∨ S (X) _E.

Theorem 4.2. Let (X, η₁, η₂, E) be a sbts. Then, every p-closed soft set is a p∨-soft set, i.e., $η_{12}^{c} \subseteq P \lor S (X)_{E}$ .

Proof. Immediate from Theorem 3.5 and Definition 4.1.

Remark 4.3. The converse of Theorem 4.2 is not true in general as shown in the following example.

Example 4.4. In Example 3.3, let $(G, E) = {(e_{1}, {x, z, w}), (e_{2}, {x, z, w})} .$ Then, ${sker}_{p}^{*} (G, E) = (G, E)$ . Hence (G, E) is a p∨-soft set but it is not p-closed soft set.

Theorem 4.5. Let (X, η₁, η₂, E) be a sbts. Then,

$(\tilde{φ}, E)$ , $(\tilde{X}, E)$ are p∨-soft sets.

An arbitrary intersection of a p∨-soft sets is a p∨-soft set.

An arbitrary union of a p∨-soft sets is a p∨-soft set.

Proof. (1) is obvious from Theorem 3.5(1) and Definition 4.1.

(2) Let {(F_i, E) : i ∈ Δ} ⊆ P ∨ S (X) _E. Then ${sker}_{p}^{*} (F_{i}, E) = (F_{i}, E)$ , ∀ i ∈ Δ, therefore ${⋂^{˜}}_{i \in Δ} {sker}_{p}^{*} (F_{i}, E) = {⋂^{˜}}_{i \in Δ} (F_{i}, E)$ . Hence, ${sker}_{p}^{*} [{⋂^{˜}}_{i \in Δ} (F_{i}, E)] = {⋂^{˜}}_{i \in Δ} (F_{i}, E)$ [by Theorem 3.5(7)]. Consequently, ${⋂^{˜}}_{i \in Δ} (F_{i}, E) \in P \lor S (X)_{E}$ .

(3) Let {(F_i, E) : i ∈ Δ} ⊆ P ∨ S (X) _E. Then, ${sker}_{p}^{*} (F_{i}, E) = (F_{i}, E)$ , ∀ i ∈ Δ. From Theorem3.5(6), we have ${⋃^{˜}}_{i \in Δ} {sker}_{p}^{*} (F_{i}, E) \tilde{\subseteq} {sker}_{p}^{*} [{⋃^{˜}}_{i \in Δ} (F_{i}, E)]$ it follows that ${⋃^{˜}}_{i \in Δ} (F_{i}, E) \tilde{\subseteq} {sker}_{p}^{*} [{⋃^{˜}}_{i \in Δ} (F_{i}, E)]$ . From Theorem 3.5(2), we have ${sker}_{p}^{*} [{⋃^{˜}}_{i \in Δ} (F_{i}, E)] \tilde{\subseteq} {⋃^{˜}}_{i \in Δ} (F_{i}, E)$ .

Consequently, ${sker}_{p}^{*} [{⋃^{˜}}_{i \in Δ} (F_{i}, E)] = {⋃^{˜}}_{i \in Δ} (F_{i}, E)$ . Hence, ${⋃^{˜}}_{i \in Δ} (F_{i}, E)$ is a p∨-soft set.

Corollary 4.6. Let (X, η₁, η₂, E) be a sbts. The family of all p∨-soft sets is a soft topology on X. This soft topology we denoted by η_p∨, i.e., $η_{p \lor} = {(G, E) \in SS (X)_{E} : {sker}_{p}^{*} (G, E) = (G, E)} .$ Moreover, (X, η_p∨, E) is an Alexandroff soft topological space.

Proof. Immediate from Theorem 4.5.

Any member of η_p∨ is called an η_p∨-open soft set and its complement is called an η_p∨-closed soft set.

Definition 4.7. The p∨-soft closure of (G, E), denoted by scl_p∨ (G, E), is the intersection of all η_p∨-closed soft supersets of (G, E), i.e., ${scl}_{p \lor} (G, E) = ⋂^{˜} {(F, E) \in η_{p \lor}^{c} : (G, E) \tilde{\subseteq} (F, E)} .$

Clearly, scl_p∨ (G, E) is the smallest η_p∨-closed soft set contains (G, E). The p∨-soft interior of (G, E), denoted by sint_p∨ (G, E), is the union of all η_p∨-soft supersets of (G, E), i.e., ${sint}_{p \lor} (G, E) = ⋃^{˜} {(H, E) \in η_{p \lor} : (H, E) \tilde{\subseteq} (G, E)} .$

Clearly, sint_p∨ (G, E) is the largest η_p∨-open soft set contained in (G, E).

Example 4.8. Let X = {x, y}, E = {e₁, e₂} and $\begin{matrix} η_{1} = {(\tilde{φ}, E), (\tilde{X}, E), (G_{1}, E), (G_{2}, E), (G_{3}, E)}, \\ η_{2} = {(\tilde{φ}, E), (\tilde{X}, E), (H_{1}, E), (H_{2}, E)}, \end{matrix}$ where $\begin{matrix} (G_{1}, E) & = & {(e_{1}, {x}), (e_{2}, φ)}, \\ (G_{2}, E) & = & {(e_{1}, {y}), (e_{2}, {y})}, \\ (G_{3}, E) & = & {(e_{1}, X), (e_{2}, {y})}, \\ (H_{1}, E) & = & {(e_{1}, {y}), (e_{2}, {x})}, \\ (H_{2}, E) & = & {(e_{1}, {x}), (e_{2}, {y})} . \end{matrix}$ Then, (X, η₁, η₂, E) is a sbts. It is easy to verify that $\begin{matrix} η_{12} & = & {(\tilde{φ}, E), (\tilde{X}, E), (G_{1}, E), (G_{2}, E), (G_{3}, E), \\ (H_{1}, E), (H_{2}, E), (P_{1}, E), (P_{2}, E)}, \end{matrix}$ where $\begin{matrix} (P_{1}, E) & = & {(e_{1}, X), (e_{2}, {x})}, \\ (P_{2}, E) & = & {(e_{1}, {y}), (e_{2}, X)} . \end{matrix}$ Therefore, $\begin{matrix} η_{12}^{c} & = & {(\tilde{φ}, E), (\tilde{X}, E), (G_{1}, E)^{c}, (G_{2}, E)^{c}, (G_{3}, E)^{c}, \\ (H_{1}, E)^{c}, (H_{2}, E)^{c}, (P_{1}, E)^{c}, (P_{2}, E)^{c}}, \end{matrix}$ where $\begin{matrix} (G_{1}, E)^{c} & = & {(e_{1}, {y}), (e_{2}, X)}, \\ (G_{2}, E)^{c} & = & {(e_{1}, {x}), (e_{2}, {x})}, \\ (G_{3}, E)^{c} & = & {(e_{1}, φ), (e_{2}, {x})}, \\ (H_{1}, E)^{c} & = & {(e_{1}, {x}), (e_{2}, {y})}, \\ (H_{2}, E)^{c} & = & {(e_{1}, {y}), (e_{2}, {x})}, \\ (P_{1}, E)^{c} & = & {(e_{1}, φ), (e_{2}, {y})}, \\ (P_{2}, E)^{c} & = & {(e_{1}, {x}), (e_{2}, φ)} . \end{matrix}$

For every $(F, E) \in η_{12}^{c}$ , we have ${sker}_{p}^{*} (F, E) = (F, E)$ [by Theorem 3.5(4)]. Since ∣SS (X) _E ∣ =2^{∣X∣.∣E∣}, we have ∣SS (X) _E ∣ =16. Therefore, $SS (X)_{E} ∖ η_{12}^{c} = {(M_{1}, E), (M_{2}, E), (M_{3}, E), (M_{4}, E), (M_{5}, E), (M_{6}, E), (M_{7}, E)}$ where $\begin{matrix} (M_{1}, E) & = & {(e_{1}, φ), (e_{2}, X)} \\ \Rightarrow {sker}_{p}^{*} (M_{1}, E) = (M_{1}, E), \\ (M_{2}, E) & = & {(e_{1}, {x}), (e_{2}, X)} \\ \Rightarrow {sker}_{p}^{*} (M_{2}, E) = (M_{2}, E), \\ (M_{3}, E) & = & {(e_{1}, {y}), (e_{2}, φ)} \\ \Rightarrow {sker}_{p}^{*} (M_{3}, E) = (\tilde{φ}, E) \neq (M_{3}, E), \\ (M_{4}, E) & = & {(e_{1}, {y}), (e_{2}, {y})} \Rightarrow {sker}_{p}^{*} (M_{4}, E) \\ = & {(e_{1}, φ), (e_{2}, {y})} \neq (M_{4}, E), \\ (M_{5}, E) & = & {(e_{1}, X), (e_{2}, φ)} \Rightarrow {sker}_{p}^{*} (M_{5}, E) \\ = & {(e_{1}, {x}), (e_{2}, φ)} \neq (M_{5}, E), \\ (M_{6}, E) & = & {(e_{1}, X), (e_{2}, {x})} \\ \Rightarrow {sker}_{p}^{*} (M_{6}, E) = (M_{6}, E), \\ (M_{7}, E) & = & {(e_{1}, X), (e_{2}, {y})} \Rightarrow {sker}_{p}^{*} (M_{7}, E) \\ = & {(e_{1}, {x}), (e_{2}, {y})} \neq (M_{7}, E) . \end{matrix}$

Consequently, $η_{p \lor} = η_{12}^{c} ⋃ {(M_{1}, E), (M_{2}, E), (M_{6}, E)}$ which is a soft topology on X. It is clear that η_p∨ ≠ SS (X) _E.

Lemma 4.9. Let (X, η₁, η₂, E) be a sbts and let (G, E) ∈ SS (X) _E. Then, (G, E) is a pΛ-soft set if and only if (G, E) ^c is a p∨-soft set.

Proof. Let (G, E) be a pΛ-soft set. Then, sker_p (G, E) = (G, E) which implies that $(G, E) = ⋂^{˜} {(H, E) \in η_{12} : (G, E) \tilde{\subseteq} (H, E)}$ , it follows that $(G, E)^{c} = ⋃^{˜} {(H, E)^{c} \in η_{12}^{c} : (H, E)^{c} \tilde{\subseteq} (G, E)^{c}}$ $= {sker}_{p}^{*} (G, E)^{c}$ . Hence, (G, E) ^c is a p∨-soft set. Similarly, we can prove the inverse direction.

Theorem 4.10. For any sbts (X, η₁, η₂, E), we have $η_{p Λ} = η_{p \lor}^{c}$ .

Proof. The proof is direct from Lemma 4.9.

Theorem 4.11. Let (X, η₁, η₂, E) be a sbts and let (G, E) , (H, E) ∈ SS (X) _E. Then,

${sker}_{p}^{*} (G, E) = [{sker}_{p} (G, E)^{c}]^{c}$ .

${sker}_{p}^{*} (G, E) \tilde{\subseteq} {sker}_{p} (G, E)$ .

${sint}_{p \lor} (G, E) = {sker}_{p}^{*} (G, E)$ and scl_p∨ (G, E) = sker_p (G, E).

Proof. (1): Since ${sker}_{p}^{*} (G, E) \tilde{\subseteq} (G, E)$ , then(G, E) ^c $\tilde{\subseteq} [{sker}_{p}^{*} (G, E)]^{c}$ implies ${sker}_{p} (G, E)^{c} \tilde{\subseteq} {sker}_{p} ([{sker}_{p}^{*} (G, E)]^{c})$ , but ${sker}_{p}^{*} (G, E)$ is a p∨-soft set [by Theorem 3.5(5)], then, by Lemma 4.9, $[{sker}_{p}^{*} (G, E)]^{c}$ is a pΛ-soft set. Therefore, ${sker}_{p} ([{sker}_{p}^{*} (G, E)]^{c}) = [{sker}_{p}^{*} (G, E)]^{c}$ . Hence, ${sker}_{p} (G, E)^{c} \tilde{\subseteq} [{sker}_{p}^{*} (G, E)]^{c}$ implies ${sker}_{p}^{*} (G, E) \tilde{\subseteq} [{sker}_{p} (G, E)^{c}]^{c}$ .

Conversely, since $(G, E)^{c} \tilde{\subseteq} {sker}_{p} (G, E)^{c}$ , then $[{sker}_{p} (G, E)^{c}]^{c} \tilde{\subseteq} (G, E)$ implies ${sker}_{p}^{*} [{sker}_{p} (G, E)^{c}]^{c} \tilde{\subseteq} {sker}_{p}^{*} (G, E)$ , but sker_p (G, E) ^c is a pΛ-softset, then, by Lemma 4.9, [sker_p (G, E) ^c] ^c is a p∨-soft set. Therefore, ${sker}_{p}^{*} ([{sker}_{p} (G, E)^{c}]^{c}) = [{sker}_{p} (G, E)^{c}]^{c}$ . Hence, $[{sker}_{p} (G, E)^{c}]^{c} \tilde{\subseteq} {sker}_{p}^{*} (G, E)$ .

(2): It is obvious.

(3): Since ${sker}_{p}^{*} (G, E), {sint}_{p \lor} (G, E)$ are η_p∨-open soft sets and ${sker}_{p}^{*} (G, E) \tilde{\subseteq} (G, E)$ , ${sint}_{p \lor} (G, E) \tilde{\subseteq} (G, E)$ , then ${sker}_{p}^{*} (G, E) \tilde{\subseteq} {sint}_{p \lor} (G, E)$ [for sint_p∨ (G, E) is the largest η_p∨-open soft set contained in (G, E)].

Conversely, since ${sint}_{p \lor} (G, E) \tilde{\subseteq} (G, E)$ , then ${sker}_{p}^{*} [{sint}_{p \lor} (G, E)] \tilde{\subseteq} {sker}_{p}^{*} (G, E)$ . But sint_p∨ (G, E) ∈ η_p∨, then ${sker}_{p}^{*} [{sint}_{p \lor} (G, E)] = {sint}_{p \lor} (G, E)$ . Thus, ${sint}_{p \lor} (G, E) \tilde{\subseteq} {sker}_{p}^{*} (G, E)$ . Hence, ${sint}_{p \lor} (G, E) = {sker}_{p}^{*} (G, E)$ . Similarly, we can prove that scl_p∨ (G, E) = sker_p (G, E).

Remark 4.12. ${sker}_{p}^{*} (G, E) \neq {sker}_{p} (G, E)$ in general as shown in the following example.

Example 4.13. In Example 4.8, consider(G, E) = {(e₁, {y}) , (e₂, {y})}. Then, ${sker}_{p}^{*} (G, E) = {(e_{1}, φ), (e_{2}, {y})}$ . But, sker_p (G, E) = {(e₁, {y}) , (e₂, {y})}. It is clear that ${sker}_{p}^{*} (G, E) \neq {sker}_{p} (G, E)$ .

5 Generalized pΛ(p∨)-soft sets

Definition 5.1. Let (X, η₁, η₂, E) be a sbts. A soft set (G, E) is called a generalized pairwise Λ-soft set [briefly, gpΛ-soft set] in a sbts (X, η₁, η₂, E) if ${sker}_{p} (G, E) \tilde{\subseteq} (F, E)$ whenever $(G, E) \tilde{\subseteq} (F, E)$ and (F, E) is a p-closed soft set.

The family of all gpΛ-soft sets in a sbts (X, η₁, η₂, E) we denoted by G_pΛS (X, η₁, η₂) _E or simply G_pΛS (X) _E.

The following example shows that G_pΛS (X) _E ≠ SS (X) _E.

Example 5.2. In Example 3.3, let $\begin{matrix} (G, E) & = & {(e_{1}, {y, z, w}), (e_{2}, {y, w})} and \\ (H, E) & = & {(e_{1}, {z, w}), (e_{2}, φ)} . \end{matrix}$

Then, (G, E) is a gpΛ-soft set because the only p-closed soft set containing (G, E) is the absolute soft set $(\tilde{X}, E)$ . But, (H, E) is not a gpΛ-soft set for, ${sker}_{p} (H, E) = {(e_{1}, {z, w}), (e_{2}, {x, w})},$ (H, E) is a p-closed soft set and $(H, E) \tilde{\subseteq} (H, E)$ but ${sker}_{p} (H, E) \tilde{⊈} (H, E)$ .

Theorem 5.3. Let (X, η₁, η₂, E) be a sbts and (G, E) ∈ SS (X) _E. Then,

(G, E) is a gpΛ-soft set ⇔ ${sker}_{p} (G, E) \tilde{\subseteq} {scl}_{p} (G, E)$ .

Proof. Straightforward.

Theorem 5.4. Every pΛ-soft set is a gpΛ-soft set, i.e., η_pΛ ⊆ G_pΛS (X) _E.

Proof. Immediate.

Remark 5.5. The converse of Theorem 5.4 is not true in general, i.e., η_pΛ ≠ G_pΛS (X) _E, as shown in the following example.

Example 5.6. In Example 3.3, let (G, E) = {(e₁, {y, z, w}) , (e₂, {y, w})}. Then, ${sker}_{p} (G, E) = {(e_{1}, {y, z, w}), (e_{2}, {x, y, w})} .$

It is clear that (G, E) is not a pΛ-soft set. On the other hand, the only p-closed soft set which containing (G, E) is the absolute soft set $(\tilde{X}, E)$ and ${sker}_{p} (G, E) \tilde{\subseteq} (\tilde{X}, E)$ . Therefore, (G, E) is a gpΛ-soft set but it is not a pΛ-soft set. Hence, η_pΛ ≠ G_pΛS (X) _E.

Theorem 5.7. For any sbts (X, η₁, η₂, E). The arbitrary union of gpΛ-soft sets is a gpΛ-soft set.

Proof. Let {(G_i, E) : i ∈ I} ⊆ G_pΛS (X) _E and let ${⋃^{˜}}_{i \in I} (G_{i}, E) \tilde{\subseteq} (F, E)$ where (F, E) is a p-closed soft set. Then, $(G_{i}, E) \tilde{\subseteq} (F, E)$ for all i ∈ I which implies that ${sker}_{p} (G_{i}, E) \tilde{\subseteq} (F, E)$ for all i ∈ I. Therefore, ${⋃^{˜}}_{i \in I} {sker}_{p} (G_{i}, E) \tilde{\subseteq} (F, E)$ , it follows that, by Theorem 2.22 (7), ${sker}_{p} [{⋃^{˜}}_{i \in I} (G_{i}, E)] \tilde{\subseteq} (F, E)$ . Hence, ${⋃^{˜}}_{i \in I} (G_{i}, E)$ is a gpΛ-soft set.

Remark 5.8. The finite intersection of gpΛ-soft sets need not be a gpΛ-soft set as shown in the following example.

Example 5.9. Let X = {a, b}, E = {e₁, e₂} and $\begin{matrix} η_{1} & = & {(\tilde{φ}, E), (\tilde{X}, E), (G, E)}, \\ η_{2} & = & {(\tilde{φ}, E), (\tilde{X}, E), (H, E)}, \end{matrix}$ where $\begin{matrix} (G, E) & = & {(e_{1}, φ), (e_{2}, {a})}, \\ (H, E) & = & {(e_{1}, {a}), (e_{2}, {b})} . \end{matrix}$

Then, (X, η₁, η₂, E) is a sbts. Consequently, $η_{12} = {(\tilde{φ}, E), (\tilde{X}, E), (G, E), (H, E), (P, E)},$

where $(P, E) = {(e_{1}, {a}), (e_{2}, X)} .$

Now, since ∣SS (X) _E ∣ =2^{∣E∣.∣X∣}, then ∣SS (X) _E ∣ =16.

Hence, SS (X) _E = η₁₂ ⋃ {(M_i, E) : i = 1, . . . , 11},

where $\begin{matrix} (M_{1}, E) & = & {(e_{1}, φ), (e_{2}, {b})}, \\ (M_{2}, E) & = & {(e_{1}, φ), (e_{2}, X)}, \\ (M_{3}, E) & = & {(e_{1}, {a}), (e_{2}, φ)}, \\ (M_{4}, E) & = & {(e_{1}, {a}), (e_{2}, {a})}, \\ (M_{5}, E) & = & {(e_{1}, {b}), (e_{2}, φ)}, \\ (M_{6}, E) & = & {(e_{1}, {b}), (e_{2}, {b})}, \\ (M_{7}, E) & = & {(e_{1}, {b}), (e_{2}, {a})}, \\ (M_{8}, E) & = & {(e_{1}, {b}), (e_{2}, X)}, \\ (M_{9}, E) & = & {(e_{1}, X), (e_{2}, φ)}, \\ (M_{10}, E) & = & {(e_{1}, X), (e_{2}, {a})}, \\ (M_{11}, E) & = & {(e_{1}, X), (e_{2}, {b})} . \end{matrix}$

It is easy to verify that sker_p (M_i, E) ≠ (M_i, E), i = 1, . . . , 11. Consequently, η₁₂ = η_pΛ. Hence, $\begin{matrix} G_{p Λ} S (X)_{E} & = & η_{12} ⋃ {(M_{1}, E), (M_{2}, E), (M_{3}, E), \\ (M_{4}, E), (M_{8}, E), (M_{10}, E)} . \end{matrix}$

Also, it is clear that (M₈, E) , (M₁₀, E) are gpΛ-soft sets but $(M_{8}, E) \tilde{\cap} (M_{10}, E) = (M_{7}, E)$ is not a gpΛ-soft set.

Theorem 5.10. Let (X, η₁, η₂, E) be a sbts. If (G, E) is a gpΛ-soft set and $(G, E) \tilde{\subseteq} (H, E) \tilde{\subseteq} {sker}_{p} (G, E)$ , then (H, E) is a gpΛ-soft set.

Proof. Let (G, E) be a gpΛ-soft set and let $(G, E) \tilde{\subseteq} (H, E) \tilde{\subseteq} {sker}_{p} (G, E)$ . Let $(H, E) \tilde{\subseteq} (F, E)$ where (F, E) is a p-closed soft set. Then, $(G, E) \tilde{\subseteq} (F, E)$ implies ${sker}_{p} (G, E) \tilde{\subseteq} (F, E)$ . Now, since $(H, E) \tilde{\subseteq} {sker}_{p} (G, E)$ , then ${sker}_{p} (H, E) \tilde{\subseteq} {sker}_{p} [{sker}_{p} (G, E)] = {sker}_{p} (G, E)$ . Therefore, ${sker}_{p} (H, E) \tilde{\subseteq} (F, E)$ . Hence, (H, E) is a gpΛ-soft set.

Theorem 5.11. Let (X, η₁, η₂, E) be a sbts.

If (G, E) is a gpΛ-soft set, then sker_p (G, E) \ (G, E) contains no non-null p-open soft set.

Proof. Let (G, E) be a gpΛ-soft set. Assume that $(H, E) \tilde{\subseteq} {sker}_{p} (G, E) \ (G, E)$ where (H, E) is a p-open soft set. Then $(H, E) \tilde{\subseteq} {sker}_{p} (G, E)$ and $(G, E) \tilde{\subseteq} (H, E)^{c}$ implies ${sker}_{p} (G, E) \tilde{\subseteq} (H, E)^{c}$ [for (G, E) is a gpΛ-soft set], it follows that ${sker}_{p} (G, E) \tilde{\cap} (H, E) = (\tilde{φ}, E)$ , but $(H, E) \tilde{\subseteq} {sker}_{p} (G, E)$ , then $(H, E) = (\tilde{φ}, E)$ . This complete the proof.

Corollary 5.12. Let (X, η₁, η₂, E) be a sbts.

If (G, E) is a gpΛ-soft set and sker_p (G, E) \ (G, E) is a p-open soft set, then (G, E) is a pΛ-soft set.

Remark 5.13. The converse of Theorem 5.11 is not true in general as shown in the following example.

Example 5.14. In Example 5.2, we have sker_p (H, E) = {(e₁, {z, w}) , (e₂, {x, w})}. Therefore, ${sker}_{p} (H, E) \ (H, E) = {(e_{1}, φ), (e_{2}, {x, w})} .$ It is clear that sker_p (H, E) \ (H, E) does not contains any null p-open soft set, nevertheless (H, E) is not a gpΛ-soft set.

Theorem 5.15. For any soft point (x_e, E) in a sbts (X, η₁, η₂, E), we have $\begin{matrix} (x_{e}, E) is a p - open soft set or (x_{e}, E)^{c} \\ is a gp Λ - soft set . \end{matrix}$

Proof. Let (x_e, E) be any soft point in a sbts (X, η₁, η₂, E). If (x_e, E) is not p-open soft set, then (x_e, E) ^c is not p-closed soft set which means that the only p-closed soft superset of (x_e, E) ^c is the absolute soft set $(\tilde{X}, E)$ . Therefore, (x_e, E) ^c is a gpΛ-soft set [for ${sker}_{p} (x_{e}, E)^{c} \tilde{\subseteq} (\tilde{X}, E)$ ].

Theorem 5.16. Let (X, η₁, η₂, E) be a sbts. Then, $\begin{matrix} (X, η_{1}, η_{2}, E) is a {PST}_{\frac{1}{2}}^{*} if and only if every \\ gp Λ - soft set is a p Λ - soft set . \end{matrix}$

Proof. Suppose that (X, η₁, η₂, E) is a ${PST}_{\frac{1}{2}}^{*}$ and let (G, E) be a gpΛ-soft set. We claim that (G, E) is a pΛ-soft set. If (G, E) is not a pΛ-soft set, then sker_p (G, E) ≠ (G, E). Therefore, there exists a soft point x_e such that $x_{e} \tilde{\in} {sker}_{p} (G, E)$ but $x_{e} \tilde{\notin} (G, E)$ . Now, since $x_{e} \tilde{\notin} (G, E)$ , then $x_{e} \tilde{\in} (G, E)^{c}$ it follows that $(G, E) \tilde{\subseteq} (x_{e}, E)^{c}$ . We conclude that (x_e, E) ^c is not p-closed soft set because if (x_e, E) ^c is a p-closed soft set, then ${sker}_{p} (G, E) \tilde{\subseteq} (x_{e}, E)^{c}$ [for (G, E) is a gpΛ-soft set] which contradicts with $x_{e} \tilde{\in} {sker}_{p} (G, E)$ . Now, since (x_e, E) ^c is not p-closed soft set, then (x_e, E) is a gp-closed soft set [by Theorem 2.30]. On the other hand, we have (x_e, E) ^c is not p-open soft set because if it is a p-open soft set, then $x_{e} \tilde{\in} {sker}_{p} (G, E) = ⋂^{˜} {(H, E) \in η_{12} : (G, E) \tilde{\subseteq} (H, E)} \tilde{\subseteq} (x_{e}, E)^{c}$ , a contradiction. Hence, (x_e, E) ^c is not p-open soft set. Thus, (x_e, E) is not a p-closed soft set. Hence, (x_e, E) is a gp-closed soft set but it is not a p-closed soft set. It follows that (X, η₁, η₂, E) is not a ${PST}_{\frac{1}{2}}^{*}$ , acontradiction.

Conversely, suppose that every gpΛ-soft set is a pΛ-soft set. Assume that (X, η₁, η₂, E) is not a ${PST}_{\frac{1}{2}}^{*}$ , then there exists a soft point (x_e, E) such that (x_e, E) is neither p-open nor p-closed soft set [by Theorem 2.31]. Since (x_e, E) is not p-open soft set, then (x_e, E) ^c is not p-closed soft set. Therefore, the only p-closed soft superset of (x_e, E) ^c is the absolute soft set $(\tilde{X}, E)$ and ${sker}_{p} (x_{e}, E)^{c} \tilde{\subseteq} (\tilde{X}, E)$ . Hence, (x_e, E) ^c is a gpΛ-soft set which implies that (x_e, E) ^c is a pΛ-soft set [by hypothesis], i.e., sker_p (x_e, E) ^c = (x_e, E) ^c. On the other hand, since (x_e, E) is not p-closed soft set, then (x_e, E) ^c is not p-open soft set. Therefore, the only p-open soft superset of (x_e, E) ^c is the absolute soft set $(\tilde{X}, E)$ . Hence, ${sker}_{p} (x_{e}, E)^{c} = (\tilde{X}, E)$ which contradicts with sker_p (x_e, E) ^c = (x_e, E) ^c. Therefore, (X, η₁, η₂, E) is a ${PST}_{\frac{1}{2}}^{*}$ .

Remark 5.17. The notions of gpΛ-soft set and gp-closed soft set are independent as shown in the following example.

Example 5.18. In Example 3.3, let (G, E) = {(e₁, {y, z, w}) , (e₂, {y, w})}. Then, (G, E) is a gpΛ-soft set [see Example 5.6]. Moreover, we have ${scl}_{p} (G, E) = (\tilde{X}, E)$ . It is clear that $(G, E) \tilde{\subseteq} (P_{2}, E)$ and (P₂, E) is a p-open soft set but ${scl}_{p} (G, E) \tilde{⊈} (P_{2}, E)$ . Hence, (G, E) is not a gp-closed soft set. On the other hand, in Example 3.3, let (H, E) = {(e₁, {z, w}) , (e₂, φ)}. Then, (H, E) is a gp-closed soft set, but from Example 5.2 we have (H, E) is not a gpΛ-soft set.

Definition 5.19. Let (X, η₁, η₂, E) be a sbts. A soft set (G, E) is called a generalized pairwise ∨-soft set [briefly, gp∨-soft set] in a sbts (X, η₁, η₂, E) if its complement is a gpΛ-soft set.

The family of all gp∨-soft sets in a sbts (X, η₁, η₂, E) we denoted by G_p∨S (X, η₁, η₂) _E or simply G_p∨S (X) _E.

Remark 5.20. $G_{p \lor} S (X)_{E} = G_{p Λ}^{c} S (X)_{E}$ , where $G_{p Λ}^{c} S (X)_{E} = {(G, E)^{c} : (G, E) \in G_{p Λ} S (X)_{E}}$ .

Theorem 5.21. Let (X, η₁, η₂, E) be a sbts and let (G, E) ∈ SS (X) _E. Then, (G, E) is a gp∨-soft set ⇔ $(H, E) \tilde{\subseteq} {sker}_{p}^{*} (G, E)$ whenever $(H, E) \tilde{\subseteq} (G, E)$ and (H, E) is a p-open soft set.

Proof. Let (G, E) be a gp∨-soft set and let $(H, E) \tilde{\subseteq} (G, E)$ such that (H, E) is a p-open soft set. Then $(G, E)^{c} \tilde{\subseteq} (H, E)^{c}$ and (H, E) ^c is a p-closed soft set implies ${sker}_{p} (G, E)^{c} \tilde{\subseteq} (H, E)^{c}$ [for (G, E) ^c is a gpΛ-soft set]. It follows that $(H, E) \tilde{\subseteq} [{sker}_{p} (G, E)^{c}]^{c}$ .

Hence, $(H, E) \tilde{\subseteq} {sker}_{p}^{*} (G, E)$ [by Theorem 4.11 (1)].

Conversely, in order to prove that (G, E) is a gp∨-soft set enough prove that (G, E) ^c is a gpΛ-soft set. Let $(G, E)^{c} \tilde{\subseteq} (F, E)$ and (F, E) is a p-closed soft set. Then, $(F, E)^{c} \tilde{\subseteq} (G, E)$ and (F, E) ^c is a p-open soft set implies $(F, E)^{c} \tilde{\subseteq} {sker}_{p}^{*} (G, E)$ [by hypothesis]. Therefore, $[{sker}_{p}^{*} (G, E)]^{c} \tilde{\subseteq} (F, E)$ implies ${sker}_{p} (G, E)^{c} \tilde{\subseteq} (F, E)$ . Hence, (G, E) ^c is gpΛ-soft set. Consequently, (G, E) is a gp∨-soft set.

Theorem 5.22. Let (X, η₁, η₂, E) be a sbts and let (G, E) ∈ SS (X) _E. Then, (G, E) is a gp∨-soft set ⇔ ${sint}_{p} (G, E) \tilde{\subseteq} {sker}_{p}^{*} (G, E)$ .

Proof. Similar to the proof of Theorem 5.3.

Theorem 5.23. Let (X, η₁, η₂, E) be a sbts. Then, (X, η₁, η₂, E) is a ${PST}_{\frac{1}{2}}^{*}$ if and only if every gp∨-soft set is a p∨-soft set.

Proof. By similar to the proof of Theorem 5.16.

6 New soft topology induced by the family of all generalized pairwise Λ-soft sets

Definition 6.1. Let (X, η₁, η₂, E) be a sbts and let (G, E) ∈ SS (X) _E. We define the operator C_gpΛ : SS (X) _E → SS (X) _E as follows: $\begin{matrix} C_{gp Λ} (G, E) & = & ⋂^{˜} {(H, E) \in G_{p Λ} (X)_{E} : \\ (G, E) \tilde{\subseteq} (H, E)} . \end{matrix}$

Lemma 6.2. For any sbts (X, η₁, η₂, E), we have $C_{gp Λ} (G, E) \tilde{\subseteq} {sker}_{p} (G, E) for all (G, E) \in SS (X)_{E} .$

Proof. It follows from the Theorems 2.26 and 5.4.

The following theorem studies the main properties of C_gpΛ operator.

Theorem 6.3. Let (X, η₁, η₂, E) be a sbts and let (G, E) , (H, E) ∈ SS (X) _E. Then,

$C_{gp Λ} (\tilde{X}, E) = (\tilde{X}, E)$ and $C_{gp Λ} (\tilde{φ}, E) = (\tilde{φ}, E)$ .

$(G, E) \tilde{\subseteq} C_{gp Λ} (G, E)$ .

$(G, E) \tilde{\subseteq} (H, E) \Rightarrow C_{gp Λ} (G, E) \tilde{\subseteq} C_{gp Λ} (H, E)$ .

(G, E) ∈ G_pΛ (X) _E ⇒ C_gpΛ (G, E) = (G, E).

$C_{gp Λ} [{\tilde{\cup}}_{i \in Δ} (G_{i}, E)] = {\tilde{\cup}}_{i \in Δ} C_{gp Λ} (G_{i}, E)$ .

$C_{gp Λ} [⋂^{˜} {(H_{i}, E) : i \in Δ}] \tilde{\subseteq} ⋂^{˜} {C_{gp Λ} (H_{i}, E) : i \in Δ}$ .

C_gpΛ [C_gpΛ (G, E)] = C_gpΛ (G, E).

Proof. (1) and (2) are obvious.

(3): Let $(G, E) \tilde{\subseteq} (H, E)$ and let $x_{e} \tilde{\in} C_{gp Λ} (G, E)$ . We claim that $x_{e} \tilde{\in} C_{gp Λ} (H, E)$ .

If $x_{e} \tilde{\notin} C_{gp Λ} (H, E)$ , then there exists a gpΛ-soft set (M, E) such that $(H, E) \tilde{\subseteq} (M, E)$ and $x_{e} \tilde{\notin} (M, E)$ . It follows that $x_{e} \tilde{\notin} C_{gp Λ} (G, E)$ , a contradiction.

(4): It is obvious from the definition of C_gpΛ (G, E).

(5): From (3) we have, ${⋃^{˜}}_{i \in Δ} C_{gp Λ} (G_{i}, E) \tilde{\subseteq} C_{gp Λ} [{⋃^{˜}}_{i \in Δ} (G_{i}, E)]$ . Now, let $x_{e} \tilde{\notin} {\tilde{\cup}}_{i \in Δ} C_{gp Λ} (G_{i}, E)$ . Then, $x_{e} \tilde{\notin} C_{gp Λ} (G_{i}, E)$ for all i ∈ Δ. Thus, for all index i ∈ Δ there exists a gpΛ-soft set (M_i, E) such that $(G_{i}, E) \tilde{\subseteq} (M_{i}, E)$ , $x_{e} \tilde{\notin} (M_{i}, E)$ . It follows that ${\tilde{\cup}}_{i \in Δ} (G_{i}, E) \tilde{\subseteq} {\tilde{\cup}}_{i \in Δ} (M_{i}, E)$ and $x_{e} \tilde{\notin} {\tilde{\cup}}_{i \in Δ} (M_{i}, E)$ . Since (M_i, E) is a gpΛ-soft set for all i, then ${\tilde{\cup}}_{i \in Δ} (M_{i}, E)$ is a gpΛ-soft set [by Theorem 5.7]. Therefore, $x_{e} \tilde{\notin} C_{gp Λ} [{\tilde{\cup}}_{i \in Δ} (G_{i}, E)]$ . It follows that $C_{gp Λ} [{\tilde{\cup}}_{i \in Δ} (G_{i}, E)] \tilde{\subseteq} {\tilde{\cup}}_{i \in Δ} C_{gp Λ} (G_{i}, E)$ . Hence, (5) holds.

(6): It follows from (3).

(7): From (2) and (3) we have $C_{gp Λ} (G, E) \tilde{\subseteq} C_{gp Λ} [C_{gp Λ} (G, E)]$ . To prove the inverse inclusion, let $x_{e} \tilde{\notin} C_{gp Λ} (G, E)$ . Then, there exists a gpΛ-soft set (M, E) such that $(G, E) \tilde{\subseteq} (M, E)$ and $x_{e} \tilde{\notin} (M, E)$ . Since $(G, E) \tilde{\subseteq} (M, E)$ , then $C_{gp Λ} (G, E) \tilde{\subseteq} C_{gp Λ} (M, E)$ [by (3)].

Therefore, $C_{gp Λ} (G, E) \tilde{\subseteq} (M, E)$ and $x_{e} \tilde{\notin} (M, E)$ [by (4)]. Thus, $x_{e} \tilde{\notin} C_{gp Λ} [C_{gp Λ} (G, E)]$ . It follows that $C_{gp Λ} [C_{gp Λ} (G, E)] \tilde{\subseteq} C_{gp Λ} (G, E)$ . Hence, (7) holds.

Theorem 6.4. C_gpΛ is a soft closure operator and induce, in usual manner, an Alexandroff soft topology that is η_gpΛ = {(G, E) ∈ SS (X) _E : C_gpΛ (G, E) ^c = (G, E) ^c}.

Proof. By Theorem 6.3, it is clear that C_gpΛ is a soft closure operator. Consequently, it is induce, in usual manner, a soft topology that is η_gpΛ = {(G, E) ∈ SS (X) _E : C_gpΛ (G, E) ^c = (G, E) ^c}. For an alexandroffness property, let {(G_i, E) : i ∈ Δ} ⊆ η_gpΛ. Then, C_gpΛ (G_i, E) ^c = (G_i, E) ^c for all i ∈ Δ. It follows that $\begin{matrix} C_{gp Λ} [{⋂^{˜}}_{i \in Δ} (G_{i}, E)]^{c} = C_{gp Λ} [{⋃^{˜}}_{i \in Δ} (G_{i}, E)^{c}] \\ = {⋃^{˜}}_{i \in Δ} C_{gp Λ} (G_{i}, E)^{c} [by Theorem 6.3 (5)] \\ = {⋃^{˜}}_{i \in Δ} (G_{i}, E)^{c} \\ = [{⋂^{˜}}_{i \in Δ} (G_{i}, E)]^{c} . \end{matrix}$ Hence, ${⋂^{˜}}_{i \in Δ} (G_{i}, E)] \in η_{gp Λ}$ .

Theorem 6.5. In any sbts (X, η₁, η₂, E), we have $η_{12}^{c} \subseteq η_{p \lor} \subseteq G_{p \lor} (X)_{E} \subseteq η_{gp Λ} \subseteq SS (X)_{E} .$

Proof. It follows from Theorems 5.4 and 6.3(4).

Remark 6.6. The equality relations in previous theorem may did not valid, i.e. η_p∨ ≠ G_p∨ (X) _E ≠ η_gpΛ ≠ SS (X) _E, as shown in the following examples.

Example 6.7. In Example 5.9, we have η_gpΛ = G_p∨ (X) _E ⋃ {(M₇, E) ^c}. It is clear that G_p∨ (X) _E ≠ η_gpΛ ≠ SS (X) _E. Also, from Example 5.6, we have η_p∨ ≠ G_p∨ (X) _E.

Example 6.8. In Example 4.8, let (G, E) = {(e₁, {x}) , ((e₂, X)}. Then ${sker}_{p} (G, E) = {scl}_{p} (G, E) = (\tilde{X}, E)$ . Therefore, by Theorem 5.3, we have (G, E) is a gpΛ-soft set. Hence, C_gpΛ (G, E) = (G, E) [by Theorem 6.3(4)]. Consequently, (G, E) ^c ∈ η_gpΛ, but (G, E) ^c = {(e₁, {y}) , ((e₂, φ)} does not belongs to η_p∨. Consequently, η_p∨ ≠ η_gpΛ.

Theorem 6.9. Let (X, η₁, η₂, E) be a sbts. Then, (X, η_gpΛ, E) is a soft $T_{\frac{1}{2}}$ .

Proof. In fact, every soft point (x_e, E) in a sbts (X, η₁, η₂, E) either p-open soft set or it is not p-open soft set. If (x_e, E) is a p-open soft set, then (x_e, E) ^c is a p-closed soft set. It follows that, by Theorem 6.5, (x_e, E) ^c ∈ η_gpΛ. Hence, (x_e, E) is an η_gpΛ-closed soft set. If (x_e, E) is not a p-open soft set, then (x_e, E) ^c is a gpΛ-soft set [by Theorem 5.15] implies C_pgΛ (x_e, E) ^c = (x_e, E) ^c. Thus, (x_e, E) is an η_gpΛ-open soft set. Consequently, every soft point either open soft set or closed soft set in (X, η_gpΛ, E). Hence, by Theorem 2.28 we have (X, η_gpΛ, E) is a soft $T_{\frac{1}{2}}$ .

Theorem 6.10. For any sbts (X, η₁, η₂, E), we have (X, η_p∨, E) is a soft $T_{\frac{1}{2}}$ if and only if η_p∨ = η_gpΛ.

Proof. Straightforward.

7 Conclusion

In this paper, we have introduced a new soft operator on soft bitopological spaces that is pairwise soft sub kernel operator and we discussed some of its properties. Based on such soft operator, we have introduced the notion of pairwise ∨-soft sets in a soft bitopological space and we studied some of its fundamental properties. Moreover, we investigated the associated soft topology η_p∨. Additionally, we have introduced the notions of generalized pairwise Λ (∨)-soft sets. Some basic properties of these notions have investigated. Based on the family of all generalized pairwise Λ-soft sets we generated a new soft topology with special properties.

Footnotes

Acknowledgments

The authors express their sincere thanks to the reviewers for their valuable suggestions. The authors are also thankful to the editors-in-chief and managing editors for their important comments which helped to improve the presentation of the paper.

References

Ali

M.I.

, Feng

, Liu

, Min

W.K.

and Shabir

On some new operations in soft set theory, Comp Math Appl 57 (2009), 1547–1553.

Aygunoglu

and Aygun

Some notes on soft topological spaces, Neural Comput Applic 21(Suppl 1) (2012), S113–S119.

Ittanagi

B.M.

Soft bitopological spaces, International Journal of Computer Applications 107(7) (2011), 1–4.

Cagman

, Karatas

and Enginoglu

Soft topology, Comput Math Appl 62 (2011), 351–358.

El-Sheikh

S.A.

and Abd El-latif

A.M.

Decompositions of some types of supra soft sets and soft continuity, International Journal of Mathematics Trends and Technology 9(1) (2014), 37–56.

Georgioua

D.N.

and Megaritis

A.C.

Soft set theory and topology, Appl Gen Topol 15(1) (2014), 93–109.

Hazra

, Majumdar

and Samanta

S.K.

Soft topology, Fuzzy Inform Eng 4(1) (2012), 105–115.

Hussain

and Ahmad

Some properties of soft topological spaces, Comput Math Appl 62 (2011), 4058–4067.

Irfan Ali

, Shabir

and Naz

, Algebraic structures of soft sets associated with new operations, Comput Math Appl 61 (2011), 2647–2654.

10.

Kandil

, Tantawy

O.A.E.

, El-Sheikh

S.A.

and Hazza

Shawqi A.

Generalized pairwise closed soft sets and the associate pairwise soft separation axioms, South Asian J Math 6(2) (2016), 43–57.

11.

Kandil

, Tantawy

O.A.E.

, El-Sheikh

S.A.

and Hazza

Shawqi A.

Pairwise open (closed) soft sets in a soft bitopological spaces, Ann Fuzzy Math Inform 11(4) (2016), 571–588.

12.

Kannan

Soft generalized closed sets in soft topological spaces, Journal of Theoretical and Appl Inform Technology 37(1) (2012), 17–21.

13.

Maji

P.K.

, Biswas

and Roy

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

14.

Majumdar

and Samanta

S.K.

On soft mappings, Comput Math Appl 60 (2010), 2666–2672.

15.

Molodtsov

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

16.

Nazmul

Sk.

and Samanta

S.K.

Neighbourhood properties of soft topological spaces, Ann Fuzzy Math Inform 6(1) (2013), 1–15.

17.

Nazmul

Sk.

and Samanta

S.K.

Some properties of soft topologies and group soft topologies, Ann Fuzzy Math Inform 8(4) (2014), 645–661.

18.

Peyghan

, Samadi

and Tayebi

About soft topological spaces, Journal of New Results in Science 2 (2013), 60–75.

19.

Pie

and Miao

From soft sets to information systems, Granular computing, IEEE Inter Conf 2 (2005), 617–621.

20.

Shabir

and Naz

On soft toplogical spaces, Comput Math Appl 61(7) (2011), 1786–1799.

21.

Das

and Samanta

S.K.

Soft metric, Ann Fuzzy Math Inform 6 (2013), 77–94.

22.

Rong

The countabilities of soft topological spaces, International Journal of Computational and Mathematical Sciences 6 (2012), 159–162.

23.

Zorlutuna

, Akdag

, Min

W.K.

and Atmaca

Remarks on soft topological spaces, Ann Fuzzy Math Inform 3(2) (2012), 171–185.