Two new forms of ordered soft separation axioms

Abstract The goal of this work is to introduce and study two new types of ordered soft separation axioms, namely soft Ti-ordered and strong soft Ti-ordered spaces (i = 0, 1, 2, 3, 4). These two types are formulated with respect to the ordinary points and the distinction between them is attributed to the nature of the monotone neighborhoods. We provide several examples to elucidate the relationships among these concepts and to show the relationships associate them with their parametric topological ordered spaces and p-soft Ti-ordered spaces. Some open problems on the relationships between strong soft Ti-ordered and soft Ti-ordered spaces (i = 2, 3, 4) are posed. Also, we prove some significant results which associate both types of the introduced ordered axioms with some notions such as finite product soft spaces, soft topological and soft hereditary properties. Furthermore, we describe the shape of increasing (decreasing) soft closed and open subsets of soft regularly ordered spaces; and demonstrate that a condition of strong soft regularly ordered is sufficient for the equivalence between p-soft T1-ordered and strong soft T1-ordered spaces. Finally, we establish a number of findings that associate soft compactness with some ordered soft separation axioms initiated in this work.


Introduction
The study of the concept of topological ordered spaces was presented for the first time by Nachbin [1]. He has constructed this concept by adding a partial order relation to the structure of a topological space. With regard to Nachbin's definition of topological ordered spaces, two points can be considered, the first one is that the topology and the partial order relation operate independently of one another, and the second one is that the topological ordered spaces are one of the generalizations of topological spaces. After Nachbin's work, many researchers carried out various studies on ordered spaces (see, for example, [2][3][4][5]).
Zadeh [6] introduced the notion of fuzzy sets in 1965 as mathematical instruments for dealing with uncertainties. To put a topological structure to fuzzy set theory, Chang [7] has defined fuzzy topological spaces. Then Katsaras [8] combined a partial order relation and a fuzzy topology to define a fuzzy topological ordered space.
In 1999, the notion of soft sets was proposed by Molodtsov [9] to overcome problems associated with uncertainties, vagueness, impreciseness and incomplete data. This notion includes enough parameters which make it a suitable alternative for the previous mathematical approaches such as fuzzy and rough sets. The useful applications of soft sets to several directions contribute to progress work on it rapidly (see, for example, [10,11]). The concept of soft topological spaces was introduced by Shabir and Naz in their pioneer work [12]. Then many studies on soft topological spaces have been done (see, for example, [13][14][15][16][17][18]). El-Shafei et al. [19] introduced partial belong and total non-belong relations which are more functional and flexible for theoretical and application studies via the soft set theory and soft topologies. Then they employed these two new notions to present new soft separation axioms, namely p-soft T i -spaces (i = 0, 1,2,3,4). The authors of [20][21][22][23][24][25] have done some amendments for some alleged results on soft axioms. Al-shami and Kočinac [26] explored the equivalence between the extended and enriched soft topologies and has obtained some interesting results related to the parametric topologies. The authors of [27,28] introduced different types of soft axioms on supra soft topological spaces.
In [29], the authors formulated the concepts of monotone soft sets and soft topological ordered spaces as a new soft structure. They also have utilized the natural belong and total non-belong relations to introduce the notions of p-soft T i -ordered spaces (i = 0, 1,2,3,4). In [30] we studied and investigated these notions on supra soft topological ordered spaces.
The topic of soft separation axioms is one of the most significant and interesting in soft topology. In general, soft separation axioms are utilized to obtain more restricted families of soft topological spaces. It turns out, from the previous studies, that there are many points of view to study soft separation axioms. The diversity of these perspectives is attributed to the relations of belong and non-belong that are used in the definitions; and the objects of study, ordinary points or soft points (see, for example, [12,19,[31][32][33][34]). The variety of ordered soft separation axioms will be more extended, because the soft neighborhoods and soft open sets is distinguished according to the partially ordered soft set.
As a contribution of study ordered soft separation axioms, the authors devote this work to defining and investigating two types of ordered soft separation axioms, namely soft T i -ordered and strong soft T i -ordered spaces (i = 0, 1, 2, 3,4). With the help of examples, we illustrate the relationships among them. Also, we derive their fundamental features such as the finite product of soft T i -ordered (resp. strong soft T i -ordered) spaces is soft T i -ordered (resp. strong soft T i -ordered) for i = 0, 1, 2; and the property of being a soft T iordered (strong soft T i -ordered) space is a soft topological ordered property for i = 0, 1, 2, 3, 4. Moreover, we investigate certain properties of them that associated with some notions of soft ordered topology such as soft ordered topological invariant and soft compatibly ordered subspaces. In the end of both Section (3) and Section (4), we discuss some results about the relationships between soft compact spaces and some of the initiated ordered soft separation axioms.

Preliminaries
This section is allocated to recall some definitions and well known results which we shall utilize them in the next parts of this work.

Soft set
Definition 2.1. [9] A pair (G, E) is said to be a soft set over X provided that G is a mapping of a parameters set E into 2 X .
For short, we use the notation G E instead of (G, E) and we express a soft set G E as follows: G E = {(e, G(e)) : e ∈ E and G(e) ∈ 2 X }. Also, we use the notation S(X E ) to denote the collection of all soft sets defined over X under a set of parameters E. Definition 2.2. [12,19] For a soft set G E over X and x ∈ X, we say that:

Definition 2.10. [37] Let G A and H B be two soft sets over X and Y, respectively. Then the cartesian product of G A and H B is denoted by (G × H) A×B and is defined as
Definition 2.11. [35] A soft mapping f ϕ of S(X A ) into S(Y B ) is a pair of mappings f : X → Y and ϕ : A → B such that for soft subsets G K and H L of S(X A ) and S(Y B ), respectively, we have: Definition 2.12. [35] A soft mapping f ϕ : S(X A ) → S(Y B ) is said to be injective (resp. surjective, bijective) if the two mappings f and ϕ are injective (resp. surjective, bijective).
Definition 2.20. [29] A soft map f ϕ : is said to be: ); (iii) an ordered embedding provided that P x α ⪯ 1 P y α if and only if f ϕ (P x α ) ⪯ 2 f ϕ (P y α ).

Soft topology
The notation τe, which is given in the proposition above, is said to be a parametric topology and (X, τe) is said to be a parametric topological space. Proposition 2.32. [29] In (X, τ, E, ⪯) we find that for each e ∈ E, the family τe = {G(e) : G E ∈ τ} with a partial order relation ⪯ form an ordered topology on X.

Ordered soft separation axioms
In this section, we formulate the concepts of soft T i -ordered spaces (i = 0, 1, 2, 3, 4) by using monotone soft neighborhoods and establish some of their properties. With the help of illustrative examples, we elucidate the relationship between them; and the interrelations between them and their parametric topological ordered spaces. (iii) an increasing (resp. a decreasing) partially soft neighborhood of x ∈ X provided that W E is an increasing (resp. a decreasing) and partially soft neighborhood of x.
The following example illustrates the above definition.  [29] reports that for every x y in X, there exist two disjoint soft neighborhoods W E and V E containing x and y, respectively. This means that y ̸ W E and x ̸ V E . [29] are still valid for soft T 2 -ordered spaces.

Since W E and V E are disjoint then y ̸ W E if and only if y ∉ W E and x ̸ V E if and only if x ∉ V E . So the definitions of soft T 2 -ordered and p-soft T 2 -ordered spaces are equivalent. Hence, all results concerning p-soft T 2 -ordered spaces in
Proof. The proof follows immediately from Definition (3.3).
To show that the converse of the above proposition is not always true, we give the following two examples. Then τ = {̃︀ Φ,̃︀ X, G i E : i = 1, 2, 3, 4} forms a soft topology on X. Now, for y x and y z, we find that W E = {(e 1 , {y}), (e 2 , X)} is an increasing soft neighborhood of y such that x ∉ W E and z ∉ W E . Also, for z x and z y, we find that W E = {(e 1 , {z}), (e 2 , X)} is an increasing soft neighborhood of z such that x ∉ W E and y ∉ W E . Therefore (X, τ, E, ⪯) is a lower soft T 1 -ordered space. Hence, it is soft T 0 -ordered. On the other hand, there does not exist a soft neighborhood W E of x such that y ∉ W E or z ∉ W E . This means that it is not an upper forms a soft topology on X. Now, for 3 2 and 3 1, we find that G 5 E is an increasing soft neighborhood of 3 such that 2 ∉ G 5 E and 1 ∉ G 5 E ; and G 4 E is a decreasing soft neighborhood of 2 and 1 such that Proof. Let a be the smallest element in (X, ⪯). Then a ⪯ x for all x ∈ X. Since ⪯ is anti-symmetric, then x ⪯̸ a for all x ∈ X. Therefore, there exists a decreasing soft neighborhood W E of a such that x ∉ W E . Since X is finite, theñ︀ ⋂︀ W E is a decreasing soft neighborhood of a such that y ∉ W E for each y ∈ X \ {a}.

Proposition 3.9. If a is the largest element of a finite lower soft T
Proof. The proof is similar to that of Proposition (3.8).

Example 3.15. Consider a a partial order relation
To prove the proposition in the case of i = 1, let (Y , τ Y , E, ⪯ Y ) be a soft ordered subspace of a soft T 1ordered space (X, τ, E, ⪯). For every a ⪯̸ Y b ∈ Y, we have a ⪯̸ b. Therefore, there is an increasing soft neighborhood W E of a and a decreasing soft neighborhood The proof in the case of i = 0 can be done similarly. Proof. The proof is complete by observing that x ̸ G E implies that x ∉ G E for every G Ẽ︀ ⊆̃︀ X.
Then there are x ∈ X and e ∈ E such that x ∈ V(e) and x ∈ d((V) c (e)). This implies that there is y ∈ (V) c (e) such that x ⪯ y. This means that y ∈ V(e). But this contradicts the disjointness of

Proposition 3.23. Every increasing (decreasing) soft closed or soft open subset of a soft regularly ordered space
Proof. Without loss of generality, suppose that H E is an increasing soft closed set in a soft regularly ordered space (X, τ, E, ⪯) which is not stable. Then there exists x ∈ X and α, β ∈ E such that x ∈ H(α) and x ∉ H(β). This means that x ∉ H E . So for any soft neighborhood W E of x and any soft neighborhood V E of H E , we obtain that x ∈ W(α) ⋂︀ V(α). Thus, we cannot find disjoint soft neighborhoods of x and H E . This is a contradiction with soft regularly ordered of (X, τ, E, ⪯). Hence, H E must be stable.
The proof of the decreasing case can be done similarly.

Corollary 3.24. If all increasing (decreasing) soft closed or soft open subsets in (X, τ, E, ⪯) are stable, then (X, τ, E, ⪯) is p-soft regularly ordered if and only if it is soft regularly ordered.
Proposition 3.25. Every soft regularly ordered space is p-soft regularly ordered.
The example below shows that the converse of Proposition (3.25) does not hold in general.

(i) The given STOS in Example(3.26) is soft T 2 -ordered and soft T 4 -ordered, but it is not soft T 3 -ordered;
(ii) If we consider (X, τ, E, ⪯) is STOS such that E is a singleton set, then (X, τ, E, ⪯) is a topological ordered space. So Example 7 in [2] shows that a soft T 4 -ordered space is a proper extension of a soft T 3 -ordered space.
The following two problems are still open.

Problem 3.28. Is a soft T 3 -ordered space a soft T 2 -ordered space?
Problem 3.29. Is a soft T 3 -ordered space a p-soft T 3 -ordered space?
The converse of Proposition 3.30 fails. We show this in the next example.

., 6} is a soft topology on X. It can be easily verified that (X, τ, E, ⪯) is soft T 4 -ordered. In contrast, we cannot find a soft open set containing y such that x does not totally belong to it. Therefore, (X, τ, E, ⪯) fails to satisfy a condition of a p-soft T 1 -ordered space. Thus, (X, τ, E, ⪯) is not p-soft T 4 -ordered.
Theorem 3.32. Every soft compatibly ordered subspace (Y , τ Y , E, ⪯ Y ) of a soft regularly ordered space (X, τ, E, ⪯) is soft regularly ordered.
is upper soft regularly ordered. In a similar manner it can be proved that (Y , τ Y , E, ⪯ Y ) is lower soft regularly ordered. Hence, (Y , τ Y , E, ⪯ Y ) is soft regularly ordered.

Corollary 3.33. Every soft compatibly ordered subspace
The proof of the next proposition is easy and thus it is omitted.

Theorem 3.35. The finite product of soft T i -ordered spaces is soft T i -ordered for i = 0, 1, 2, 3.
Proof. We only prove the theorem in the case of i = 2, and the other cases can be proved similarly.
Assume that (X, τ 1 , E 1 , ⪯ 1 ) and (Y , τ 2 , E 2 , ⪯ 2 ) are soft T 2 -ordered spaces and let (X × Y , τ, E, ⪯) be the soft ordered product space of them. Let (x 1 , y 1 ) ⪯̸ (x 2 , y 2 ) ∈ X × Y. Then x 1 ⪯̸ 1 x 2 or y 1 ⪯̸ 2 y 2 . Without loss of generality, say x 1 ⪯̸ 1 x 2 . Since (X, τ 1 , E 1 , ⪯ 1 ) is soft T 2 -ordered, then there is an increasing soft neighborhood W E1 of x 1 and a decreasing soft neighborhood V E1 of x 2 such that x 2 ∉ W E1 and x 1 ∉ V E1 which are disjoint. Therefore, W E1 ×̃︀ Y is an increasing soft neighborhood of (x 1 , y 1 ) and V E1 ×̃︀ Y is a decreasing soft neighborhood of (x 2 , y 2 ) such that (x 2 , Proof. We only prove the theorem in the cases of i = 2, 4, and the other cases can be proved similarly. (i) Let f ϕ : (X, τ, A, ⪯ 1 ) → (Y , θ, B, ⪯ 2 ) be an ordered embedding soft homeomorphism map such that (X, τ, A, ⪯ 1 ) is soft T 2 -ordered. Suppose that x ⪯̸ 2 y ∈ Y. Then P x β ⪯̸ 2 P y β for each β ∈ B. Since f ϕ is bijective, then there are P a α and P b α iñ︀ X such that f ϕ (P a α ) = P x β and f ϕ (P b α ) = P y β and since f ϕ is an ordered embedding, then P a α ⪯̸ 1 P b α . So a ⪯̸ 1 b. By hypothesis, we have an increasing soft neighborhood V E of a and a decreasing soft neighborhood Since f ϕ is bijective soft open, then f ϕ (V E ) and f ϕ (W E ) are disjoint soft neighborhoods of x and y, respectively. From Theorem (2.21), we obtain f ϕ (V E ) and f ϕ (W E ) are increasing and decreasing, respectively. Hence, the proof is complete.
(ii) Let f ϕ : (X, τ, A, ⪯ 1 ) → (Y , θ, B A similar proof can be given for decreasing case. Proof. Suppose that F 1 E and F 2 E are two disjoint soft closed sets such that F 1 E is decreasing and F 2 E is increasing. Then F 2 Ẽ︀ ⊆F c 1 E . Since (X, τ, E, ⪯) is soft compact, then F 2 E is soft compact and since (X, τ, E, ⪯) is soft regularly ordered, then there is an increasing soft neighborhood suppose that there exists an element x ∈ G E and x ∈ d[(G E ) c ]. So there exists an element y ∈ (G E ) c such that x ⪯ y. This means that y ∈ G E . But this contradicts the disjointness of G E and (G E ) c . Thus, (X, τ, E, ⪯) is soft normally ordered.
To show that the converse of the above theorem and corollary fail we give the following example. Obviously, (X, τ, E, ⪯) is soft normally ordered and soft compact. Also, for every increasing (resp. decreasing) soft compact subset of (X, τ, E, ⪯) and every decreasing (

Strong ordered soft separation axioms
The first aim of this section is to define strong ordered soft separation axioms, namely strong soft T i -ordered spaces (i = 0, 1, 2, 3, 4) by using monotone soft open sets in the place of monotone soft neighborhoods. The second aim is to provide some examples to illustrate the relationships between these and the relationships between them and soft T i -ordered spaces. The third aim is to discuss their main properties and provide some results that associate soft compactness and some initiated strong ordered soft separation axioms.
The following example explains the difference between soft open sets and soft neighborhoods in terms of increasing and decreasing.

(i) Every monotone soft open set containing an element x is a monotone soft neighborhood of x. (ii) Every monotone soft open set containing a soft set H E is a monotone soft neighborhood of H E .
Proof. Let G E be a monotone soft open set containing an element x. Then x ∈ G E ⊆ G E . Therefore, G E is a monotone soft neighborhood of x. Also, if G E is a monotone soft open set containing a soft set H E . Then H E ⊆ G E ⊆ G E . Therefore, G E is a monotone soft neighborhood of H E . Example (4.1) demonstrates that the converse of the above proposition fails.    To show that the converse of the above corollary fails, we give the following example. To prove that (iii)→(i), let x ⪯̸ y ∈ X. Since (X, τ, E, ⪯) is strong soft T 0 -ordered, then it is strong lower soft T 1 -ordered or strong upper soft T 1 -ordered. Say, it is strong upper soft T 1 -ordered. It follows, by the above corollary, that (i(x)) E is an increasing soft closed set. Since y ∉ (i(x)) E and (X, τ, E, ⪯) is strong soft regularly ordered, then there exist disjoint soft open sets W E and V E containing (i(x)) E and y, respectively, such that W E is increasing and V E is decreasing. Hence, the proof is complete.     Proof. The proof follows directly from the definitions of strong soft T i -ordered and soft T i -spaces.

Remark 4.25.
To confirm that the converse of the above proposition fails, we consider E is a singleton and then we suffice with the examples introduced in [2]. Also, by considering E is a singleton, Example 3 in [2] shows that the concepts of strong soft T i -ordered and soft T i -spaces (i = 3, 4) are independent of each other.
In conclusion, we give Figure 1 to illustrate the relationships among some types of ordered soft separation axioms.

Conclusion and future work
By combining a partial order relation and a topology on a non-empty set, Nachbin [1] defined the topological ordered space. Similarly, Al-shami et al. [29] defined the soft topological ordered space. Studying soft separation axioms via soft topological spaces is a significant topic because they help establish a wider family which can be easily applied to classify the objects under study. We demonstrate in the last paragraph of introduction the reasons for doing many studies via soft separation axioms and the variety of these studies will be more via ordered soft separation axioms. Throughout this work, we use the notions of monotone soft neighborhoods and monotone soft open sets to present soft T i -ordered and strong soft T i -ordered spaces, respectively, for i = 0, 1, 2, 3, 4. These two types are formulated with respect to the ordinary points. We establish several results such as strong soft T i -ordered spaces is strictly finer than soft T i -ordered spaces and support this result with number of interesting examples. Also, we discuss the relationships which associate the soft T i -ordered (strong soft T i -ordered) spaces with p-soft T i -ordered spaces and soft T i -spaces. In Theorem (4.8), we give a condition that satisfies the equivalence between p-soft T 1 -ordered and strong soft T 1 -ordered spaces. In the end of Section (3) and Section (4), we present a number of results that associate soft compactness with some of the initiated ordered soft separation axioms. Some open problems on the relationship between strong soft T i -ordered and soft T i -ordered spaces (i = 2, 3, 4) are posed.
To extend this study, one can generalize the initiated concepts on supra soft topological spaces [40]. All these results will provide a base to researchers who want to work in the soft ordered topology field and will help to establish a general framework for applications in practical fields.