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 ηp∨. 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 ηgpΛ on X which is finer than ηp∨. Furthermore, we prove that (X, ηgpΛ, E) is always a soft space. In addition, we introduce characterizations of pairwise soft 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.
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 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 ηpΛ 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, and .
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 ηgpΛ which is finer than ηp∨. Furthermore, we prove that (X, ηpgΛ, E) is always a soft space. In addition, we introduce the characterization of pairwise soft by using generalized pairwise Λ-soft sets.
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
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 , 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 , 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 , 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 or , if F (e) = φ, ∀e ∈ E.
A soft set (F, E) over X is said to be an absolute soft set, denoted by or , 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 = (Fc, E), where Fc : E → P (X) is a mapping given by Fc (e) = X \ F (e), ∀ e ∈ E.
Clearly, (Fc) c is the same as F, ((F, E) c) c = (F, E), and .
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, .
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 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 (xe, E) or xe, i.e., xe : E → P (X) is a mapping defined by
The set of all soft points in is denoted by ξ (X) E.
Definition 2.6. [21] The soft point (xe, E) is said to be belonging to the soft set (G, E), denoted by , if xe (e) ⊆ G (e), i.e., {x} ⊆ G (e). Clearly, if and only if .
Definition 2.7. [21] The two soft points xe1, ye2 over X are said to be equal if x = y and e1 = e2. Thus, xe1 ≠ ye2 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., .
Proposition 2.9.[21] Let (G, E), (H, E) be two soft sets over X. Then,
.
or .
and .
.
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:
,
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 ηc.
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 sclη (F, E), is the intersection of all closed soft super sets of (F, E), i.e, . Clearly sclη (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,
∀ (O, E) ∈ η (xe), whereη (xe) denote the family of all open soft sets which containsxe.
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 whenever and (H, E) ∈ η.
Definition 2.14. [12] A a soft topological space (X, η, E) is called a soft [briefly, ] 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:
, ,
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 . 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 .
The family of all p-open (p-closed) soft sets in sbts (X, η1, η2, E) is denoted by η12 (), 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
. 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 sclp (G, E), is the intersection of all p-closed soft supersets of (G, E), i.e.,
Clearly, sclp (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 sintp (G, E), is the union of all p-open soft subsets of (G, E), i.e.,
Clearly, sintp (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, skerp (G, E)], is the intersection of all p-open soft supersets of (G, E), i.e.,
Theorem 2.22.[11] Let (X, η1, η2, E) be a sbts and let (G, E) , (H, E) ∈ SS (X) E. Then,
and .
.
.
(G, E) ∈ η12 ⇒ skerp (G, E) = (G, E).
skerp [skerp (G, E)] = skerp (G, E).
.
.
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 skerp (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 ηPΛ. The triple (X, ηpΛ, 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,
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 whenever and (H, E) is a p-open soft set.
Theorem 2.28.[10] A soft topological space (X, η, E) is a soft 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 [briefly, ] 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 if and only if every soft point in (X, η1, η2, E) either p-open or p-closed soft set.
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, ] is the union of all p-closed soft subsets of (G, E), i.e.,
where 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
such that
It is easy to verify that
where
Therefore, the family of all p-closed soft sets is
where
It follows that,
It is clear that 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,
such that .
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,
and .
.
.
.
.
.
.
Proof. From (1) to (4), it is obvious from Definition 3.1.
(5) Since , then . To prove the inverse inclusion, let . Then, there exists such that . Therefore, by using (3) and (4), we have . Hence, , i.e., . Therefore, (5) holds.
(6) Since ,then .Therefore, .
(7) Since , then for all i ∈ Δ implies . To prove the inverse inclusion, let . Then, ∀ i ∈ Δ. Thus, by Theorem 3.4, for all index i ∈ Δ there exists such that . We take . It follows that and . Since , then which implies that .
Therefore, . Hence, (7) holds.
Remark 3.6. 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, and. It followsthat, . On the other hand, .
It is clear that .
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 . 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., .
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
Then, . 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,
, 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 {(Fi, E) : i ∈ Δ} ⊆ P ∨ S (X) E. Then, ∀ i ∈ Δ, therefore . Hence, [by Theorem 3.5(7)]. Consequently, .
(3) Let {(Fi, E) : i ∈ Δ} ⊆ P ∨ S (X) E. Then,, ∀ i ∈ Δ. From Theorem3.5(6), we have it follows that . From Theorem 3.5(2), we have .
Consequently, . Hence, 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 ηp∨, i.e.,
Moreover, (X, ηp∨, E) is an Alexandroff soft topological space.
Proof. Immediate from Theorem 4.5.
Any member of ηp∨ is called an ηp∨-open soft set and its complement is called an ηp∨-closed soft set.
Definition 4.7. The p∨-soft closure of (G, E), denoted by sclp∨ (G, E), is the intersection of all ηp∨-closed soft supersets of (G, E), i.e.,
Clearly, sclp∨ (G, E) is the smallest ηp∨-closed soft set contains (G, E). The p∨-soft interior of (G, E), denoted by sintp∨ (G, E), is the union of all ηp∨-soft supersets of (G, E), i.e.,
Clearly, sintp∨ (G, E) is the largest ηp∨-open soft set contained in (G, E).
Example 4.8. Let X = {x, y}, E = {e1, e2} and
where
Then, (X, η1, η2, E) is a sbts. It is easy to verify that
where
Therefore,
where
For every , we have [by Theorem 3.5(4)]. Since ∣SS (X) E ∣ =2∣X∣.∣E∣, we have ∣SS (X) E ∣ =16. Therefore, where
Consequently, which is a soft topology on X. It is clear that ηp∨ ≠ 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, skerp (G, E) = (G, E) which implies that , it follows that . 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 .
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,
.
.
andsclp∨ (G, E) = skerp (G, E).
Proof. (1): Since , then(G, E) c implies , but is a p∨-soft set [by Theorem 3.5(5)], then, by Lemma 4.9, is a pΛ-soft set. Therefore,. Hence, implies .
Conversely, since , then implies , but skerp (G, E) c is a pΛ-softset, then, by Lemma 4.9, [skerp (G, E) c] c is a p∨-soft set. Therefore, . Hence, .
(2): It is obvious.
(3): Since are ηp∨-open soft sets and , , then [for sintp∨ (G, E) is the largest ηp∨-open soft set contained in (G, E)].
Conversely, since , then. But sintp∨ (G, E) ∈ ηp∨, then . Thus, . Hence, . Similarly, we can prove that sclp∨ (G, E) = skerp (G, E).
Remark 4.12. in general as shown in the following example.
Example 4.13. In Example 4.8, consider(G, E) = {(e1, {y}) , (e2, {y})}. Then, . But, skerp (G, E) = {(e1, {y}) , (e2, {y})}. It is clear that .
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 whenever 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 GpΛS (X, η1, η2) E or simply GpΛS (X) E.
The following example shows that GpΛS (X) E ≠ SS (X) E.
Example 5.2. In Example 3.3, let
Then, (G, E) is a gpΛ-soft set because the only p-closed soft set containing (G, E) is the absolute soft set . But, (H, E) is not a gpΛ-soft set for,
(H, E) is a p-closed soft set and but .
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 ⇔ .
Proof. Straightforward.
Theorem 5.4.Every pΛ-soft set is a gpΛ-soft set, i.e., ηpΛ ⊆ GpΛS (X) E.
Proof. Immediate.
Remark 5.5. The converse of Theorem 5.4 is not true in general, i.e., ηpΛ ≠ GpΛS (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,
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 and . Therefore, (G, E) is a gpΛ-soft set but it is not a pΛ-soft set. Hence, ηpΛ ≠ GpΛS (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 {(Gi, E) : i ∈ I} ⊆ GpΛS (X) E and let where (F, E) is a p-closed soft set. Then, for all i ∈ I which implies that for all i ∈ I. Therefore, , it follows that, by Theorem 2.22 (7), . Hence, 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
where
Then, (X, η1, η2, E) is a sbts. Consequently,
where
Now, since ∣SS (X) E ∣ =2∣E∣.∣X∣, then ∣SS (X) E ∣ =16.
Hence, SS (X) E = η12 ⋃ {(Mi, E) : i = 1, . . . , 11},
where
It is easy to verify that skerp (Mi, E) ≠ (Mi, E), i = 1, . . . , 11. Consequently, η12 = ηpΛ. Hence,
Also, it is clear that (M8, E) , (M10, E) are gpΛ-soft sets but 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 , then (H, E) is a gpΛ-soft set.
Proof. Let (G, E) be a gpΛ-soft set and let . Let where (F, E) is a p-closed soft set. Then, implies . Now, since , then . Therefore, . 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 skerp (G, E) \ (G, E) contains no non-null p-open soft set.
Proof. Let (G, E) be a gpΛ-soft set. Assume that where (H, E) is a p-open soft set. Then and implies [for (G, E) is a gpΛ-soft set], it follows that, but , then . This complete the proof.
Corollary 5.12.Let (X, η1, η2, E) be a sbts.
If (G, E) is a gpΛ-soft set and skerp (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 skerp (H, E) = {(e1, {z, w}) , (e2, {x, w})}. Therefore,
It is clear that skerp (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 (xe, E) in a sbts (X, η1, η2, E), we have
Proof. Let (xe, E) be any soft point in a sbts (X, η1, η2, E). If (xe, E) is not p-open soft set, then (xe, E) c is not p-closed soft set which means that the only p-closed soft superset of (xe, E) c is the absolute soft set . Therefore, (xe, E) c is a gpΛ-soft set [for ].
Theorem 5.16.Let (X, η1, η2, E) be a sbts. Then,
Proof. Suppose that (X, η1, η2, E) is a 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 skerp (G, E) ≠ (G, E). Therefore, there exists a soft point xe such that but . Now, since , then it follows that . We conclude that (xe, E) c is not p-closed soft set because if (xe, E) c is a p-closed soft set, then [for (G, E) is a gpΛ-soft set] which contradicts with . Now, since (xe, E) c is not p-closed soft set, then (xe, E) is a gp-closed soft set [by Theorem 2.30]. On the other hand, we have (xe, E) c is not p-open soft set because if it is a p-open soft set, then , a contradiction. Hence, (xe, E) c is not p-open soft set. Thus, (xe, E) is not a p-closed soft set. Hence, (xe, 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 , acontradiction.
Conversely, suppose that every gpΛ-soft set is a pΛ-soft set. Assume that (X, η1, η2, E) is not a , then there exists a soft point (xe, E) such that (xe, E) is neither p-open nor p-closed soft set [by Theorem 2.31]. Since (xe, E) is not p-open soft set, then (xe, E) c is not p-closed soft set. Therefore, the only p-closed soft superset of (xe, E) c is the absolute soft set and . Hence, (xe, E) c is a gpΛ-soft set which implies that (xe, E) c is a pΛ-soft set [by hypothesis], i.e., skerp (xe, E) c = (xe, E) c. On the other hand, since (xe, E) is not p-closed soft set, then (xe, E) c is not p-open soft set. Therefore, the only p-open soft superset of (xe, E) c is the absolute soft set . Hence, which contradicts with skerp (xe, E) c = (xe, E) c. Therefore, (X, η1, η2, E) is a .
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 . It is clear that and (P2, E) is a p-open soft set but . 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 Gp∨S (X, η1, η2) E or simply Gp∨S (X) E.
Remark 5.20. , where .
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 ⇔ whenever and (H, E) is a p-open soft set.
Proof. Let (G, E) be a gp∨-soft set and let such that (H, E) is a p-open soft set. Then and (H, E) c is a p-closed soft set implies [for (G, E) c is a gpΛ-soft set]. It follows that .
Hence, [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 and (F, E) is a p-closed soft set. Then, and (F, E) c is a p-open soft set implies [by hypothesis]. Therefore, implies . 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 ⇔ .
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 if and only if every gp∨-soft set is a p∨-soft set.
Proof. By similar to the proof of Theorem 5.16.
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 CgpΛ : SS (X) E → SS (X) E as follows:
Lemma 6.2.For any sbts (X, η1, η2, E), we have
Proof. It follows from the Theorems 2.26 and 5.4.
The following theorem studies the main properties of CgpΛ operator.
Theorem 6.3.Let (X, η1, η2, E) be a sbts and let (G, E) , (H, E) ∈ SS (X) E. Then,
and .
.
.
(G, E) ∈ GpΛ (X) E ⇒ CgpΛ (G, E) = (G, E).
.
.
CgpΛ [CgpΛ (G, E)] = CgpΛ (G, E).
Proof. (1) and (2) are obvious.
(3): Let and let . We claim that .
If , then there exists a gpΛ-soft set (M, E) such that and . It follows that , a contradiction.
(4): It is obvious from the definition of CgpΛ (G, E).
(5): From (3) we have, . Now, let . Then, for all i ∈ Δ. Thus, for all index i ∈ Δ there exists a gpΛ-soft set (Mi, E) such that , . It follows that and . Since (Mi, E) is a gpΛ-soft set for all i, then is a gpΛ-soft set [by Theorem 5.7]. Therefore, . It follows that . Hence, (5) holds.
(6): It follows from (3).
(7): From (2) and (3) we have . To prove the inverse inclusion, let . Then, there exists a gpΛ-soft set (M, E) such that and . Since , then [by (3)].
Therefore, and [by (4)]. Thus, . It follows that . Hence, (7) holds.
Theorem 6.4.CgpΛ is a soft closure operator and induce, in usual manner, an Alexandroff soft topology that is ηgpΛ = {(G, E) ∈ SS (X) E : CgpΛ (G, E) c = (G, E) c}.
Proof. By Theorem 6.3, it is clear that CgpΛ is a soft closure operator. Consequently, it is induce, in usual manner, a soft topology that is ηgpΛ = {(G, E) ∈ SS (X) E : CgpΛ (G, E) c = (G, E) c}. For an alexandroffness property, let {(Gi, E) : i ∈ Δ} ⊆ ηgpΛ. Then, CgpΛ (Gi, E) c = (Gi, E) c for all i ∈ Δ. It follows that
Hence, .
Theorem 6.5.In any sbts (X, η1, η2, E), we have
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. ηp∨ ≠ Gp∨ (X) E ≠ ηgpΛ ≠ SS (X) E, as shown in the following examples.
Example 6.7. In Example 5.9, we have ηgpΛ = Gp∨ (X) E ⋃ {(M7, E) c}. It is clear that Gp∨ (X) E ≠ ηgpΛ ≠ SS (X) E. Also, from Example 5.6, we have ηp∨ ≠ Gp∨ (X) E.
Example 6.8. In Example 4.8, let (G, E) = {(e1, {x}) , ((e2, X)}. Then . Therefore, by Theorem 5.3, we have (G, E) is a gpΛ-soft set. Hence, CgpΛ (G, E) = (G, E) [by Theorem 6.3(4)]. Consequently, (G, E) c ∈ ηgpΛ, but (G, E) c = {(e1, {y}) , ((e2, φ)} does not belongs to ηp∨. Consequently, ηp∨ ≠ ηgpΛ.
Theorem 6.9.Let (X, η1, η2, E) be a sbts. Then, (X, ηgpΛ, E) is a soft .
Proof. In fact, every soft point (xe, E) in a sbts (X, η1, η2, E) either p-open soft set or it is not p-open soft set. If (xe, E) is a p-open soft set, then (xe, E) c is a p-closed soft set. It follows that, by Theorem 6.5, (xe, E) c ∈ ηgpΛ. Hence, (xe, E) is an ηgpΛ-closed soft set. If (xe, E) is not a p-open soft set, then (xe, E) c is a gpΛ-soft set [by Theorem 5.15] implies CpgΛ (xe, E) c = (xe, E) c. Thus, (xe, E) is an ηgpΛ-open soft set. Consequently, every soft point either open soft set or closed soft set in (X, ηgpΛ, E). Hence, by Theorem 2.28 we have (X, ηgpΛ, E) is a soft .
Theorem 6.10.For any sbts (X, η1, η2, E), we have (X, ηp∨, E) is a soft if and only if ηp∨ = ηgpΛ.
Proof. Straightforward.
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 ηp∨. 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.
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