Compactness on Soft Topological Ordered Spaces and Its Application on the Information System

It is well known every soft topological space induced from soft information system is soft compact. In this study, we integrate between soft compactness and partially ordered set to introduce new types of soft compactness on the finite spaces and investigate their application on the information system. First, we initiate a notion of monotonic soft sets and establish its main properties. Second, we introduce the concepts of monotonic soft compact and ordered soft compact spaces and show the relationships between them with the help of examples. We give a complete description for each one of them by making use of the finite intersection property. Also, we study some properties associated with some soft ordered spaces and finite product spaces. Furthermore, we investigate the conditions under which these concepts are preserved between the soft topological ordered space and its parametric topological ordered spaces. In the end, we provide an algorithm for expecting the missing values of objects on the information system depending on the concept of ordered soft compact spaces.


Introduction
Compactness is a property that generalizes the notion of a closed and bounded subset of Euclidean space. It has been described by using the finite intersection property for closed sets.
e important motivations beyond studying compactness have been given in [1]. Without doubt, the concept of compactness occupied a wide area of topologists' attention. Many relevant ideas to this concept have been introduced and studied. Generalizations of compactness have been formulated in many directions, one of them given by using generalized open sets; see, for example, [2].
In 1965, Nachbin [3] combined a partial order relation with a topological space to define a new mathematical structure, namely, a topological ordered space. en, McCartan [4] formulated ordered separation axioms with respect to open sets and neighbourhoods which were described by increasing or decreasing operators. Shabir and Gupta [5] extended these ordered separation axioms in the cases of T 1 -ordered and T 2 -ordered spaces using semiopen sets. Recently, Al-Shami and Abo-Elhamayel [6] introduced new types of ordered separation axioms.
In 1999, Molotdsov [7] came up with the idea of soft sets for dealing with uncertainties and vagueness. Shabir and Naz [8], in 2011, exploited soft sets to introduce the concept of soft topological spaces and study soft separation axioms. en, researchers started working to generalize topological notions on the soft topological frame. In this regard, compactness was one of the topics that received much attention. It was presented and explored firstly by Molodtsov Ayg .. unoglu and Ayg .. un, and Zorlutuna et al. [9,10]. en, Hida [11] compared between two types of soft compactness. After that, Al-Shami et al. [12] defined almost soft compact and mildly soft compact spaces and investigated the main properties. Lack of consideration in the privacy of soft sets by some authors causes emerging some alleged results, especially those related to the properties of soft compactness. erefore, the authors of [13][14][15][16][17] carried out some corrective studies in this regard.
In 2019, Al-Shami et al. [18] defined topological ordered spaces on soft setting. ey studied monotonic soft sets and then utilized them to present p-soft T i -ordered spaces. Also, they [19] presented soft I(D, B)-continuous mappings and established several generalizations of them using generalized soft open sets. Moreover, Al-Shami and El-Shafei [20] initiated two types of ordered separation axioms, namely, soft T i -ordered and strong soft T i -ordered spaces. Recently, the concept of supra soft topological ordered spaces has been explored and discussed in [21]. is paper is organized as follows: Section 2 gives some main notions of monotonic soft sets and soft topological ordered spaces. In Section 3, we introduce the concept of monotonic soft sets and show some properties with the help of examples. Section 4 defines and explores a concept of monotonic soft compact spaces. By using a convenient technique, we show that many well-known results of soft compactness are valid on monotonic soft compact spaces. We divide Section 5 into two parts: the first one studies a concept of ordered soft compact spaces, and the second one makes use of it to present a practical application on the information system. Section 6 concludes the paper with summary and further works.

Preliminaries
To make this work self-contained, we mention some definitions and results that were introduced in soft set theory, soft topology, and ordered soft topology.

Soft Set
Definition 1 (see [7]). A soft set over X ≠ ∅ is a mapping G from a parameter set E to the power set 2 X of X. It is denoted by (G, E).
Usually, we write a soft set (G, E) as a set of ordered pairs. In other words, (G, E) � (e, G(e)): { e ∈ E and G(e) ∈ 2 X }.
Sometimes, we use some symbols such as A, B in place of E and F, H in place of G.
Definition 2 (G, E) over X is said to be (i) a null soft set (resp. an absolute soft set) [22] if G(e) � ∅ (resp. G(e) � X) for each e ∈ E; and it is denoted by Φ (resp. X). (ii) soft point [23,24] It is denoted by P x e . (iii) finite (resp. countable) soft set [23] if G(e) is finite (resp. countable) for each e ∈ E. Otherwise, it is called infinite (resp. uncountable).
Definition 3 (see [25]). e relative complement of (G, E), Definition 4 (see [26]). (G, A) is a subset of (F, B) if A⊆B and G(a)⊆F(a) for all a ∈ A.
Definition 5 (see [22]). e union of two soft sets (G, A) and , and a mapping V: D ⟶ 2 X is defined as follows: Definition 6 (see [25]). e intersection of soft sets (G, A) and (F, Definition 7 (see [9,10]). Let (G, A) and (F, B) be two soft sets over X and Y, respectively. en, the Cartesian product of (G, A) and Definition 8 (see [8,27]). For a soft set (G, A) over X and x ∈ X, we say that for each e ∈ E Proposition 1 (see [10,24]). For a soft mapping , we have the following results: Definition 9 (see [28]). A binary relation ≺ is called a partial order relation if it is reflexive, antisymmetric, and transitive. e pair (X, ≺ ) is called a partially ordered set. e relation on a nonempty set X which is given by } is called the equality relation and is denoted by △.
Definition 10 (see [18]). (X, E, ≺ ) is said to be a partially ordered soft set on X ≠ ∅ if (X, ≺ ) is a partially ordered set. For two soft points P x α and P y α in (X, E, ≺ ), we say that Definition 11 (see [18]). Increasing operator i and decreasing operator d are two maps of (S(X E ), ≺ ) into (S(X E ), ≺ ) defined as follows: for each soft subset (G, E) of S(X E ), Definition 12. Let (X, E, ≺ 1 ) and (Y, E, ≺ 2 ) be two partially ordered soft sets. e product relation ≺ of ≺ 1 and ≺ 2 on X × Y is defined as follows: Definition 13 (see [18]). A soft subset (G, E) of (X, E, ≺ ) is said to be increasing (resp. decreasing) provided that Theorem 1 (see [18]). e finite product of increasing (resp. decreasing) soft sets is increasing (resp. decreasing).
be a subjective ordered embedding soft mapping. en, the image of each increasing (resp. decreasing) soft set is increasing (resp. decreasing).

Soft Topological Space
Definition 15 (see [8]). e family τ of soft sets over X under a fixed parameter set E is said to be a soft topology on X if it contains X and Φ and is closed under a finite intersection and an arbitrary union. e triple (X, τ, E) is said to be a soft topological space. Every member of τ is called soft open, and its relative complement is called soft closed.
Many properties of extended soft topologies which help us to show the relationships between soft topology and its parametric topologies were studied in [30].

Soft Topological Ordered Space
Definition 19 (see [18]). A quadrable system (X, τ, E, ≺ ) is said to be a soft topological ordered space if (X, τ, E) is a soft topological space and (X, ≺ ) is a partially ordered set.
Definition 20 (see [18]). A soft subset (W, E) of (X, τ, E, ≺ ) is said to be increasing (resp. decreasing) soft neighborhood of x ∈ X if (W, E) is increasing (resp. decreasing) and a soft neighborhood of x.
(X, τ e , ≺ ) is said to be a parametric topological ordered space.
Definition 22 (see [18]). e product of a finite family of soft topological ordered spaces Definition 23 (see [18]). A soft ordered subspace (Y, τ Y , ≺ Y , E) of (X, τ, E, ≺ ) is called soft compatibly ordered if for each increasing (resp. decreasing) soft closed subset (H, E) of (Y, τ Y , ≺ Y , E), there exists an increasing (resp. a decreasing) soft closed subset Definition 24 (see [18] If the phrase "soft neighborhoods" is replaced by "soft open sets," then the above soft axioms are called strong psoft regularly ordered and strong p-soft T i -ordered spaces, i � 2, 3.

Monotonically Soft Sets
In this section, we introduce a concept of monotonic soft sets and study the main properties with the help of illustrative examples.
e following example shows that the union and intersection of monotonic soft sets are not always monotonic.
{ } be a partial order relation on X � 1, 2, 3, 4, 5 { }, and let E � e 1 , e 2 be a set of parameters. We define the following three subsets (F, E), (G, E), and (H, E) of (X, E, ≺ ) as follows: Since e following result demonstrates under what conditions the union (intersection) of monotonic soft sets is monotonic. □ Corollary 1. Let (F, E) and (G, E) be two monotonic subsets of (X, E, ≺ 1 ) and (Y, E, ≺ 2 ), respectively. en, we have the following two results: , respectively. We have the following two results: Proof. We only prove (i), and one can prove (ii) similarly.
□ e next example illustrates that the product of two monotonic soft sets need not be a monotonic soft set.
{ } is given as follows: e following result demonstrates under what conditions the product of two monotonic soft sets is a monotonic soft set. Proposition 6. Let (F, E) and (G, E) be two subsets of (X, E, ≺ 1 ) and (Y, E, ≺ 2 ), respectively, such that (F, E) and (G, E) are both increasing (decreasing). en, (F, E) and Proof. It follows immediately from Proposition 5.
In view of eorem 2, we obtain the following result. □

Monotonically Soft Compact Spaces
In this section, we introduce a concept of monotonic soft compact spaces and study their main properties. Also, we characterize them and demonstrate their relationships with ST i -ordered spaces (i � 2, 3, 4). Finally, we investigate some results that associated the concept of monotonic compact spaces with monotonic limit points and some types of monotonic maps.
is said to be monotonic soft compact provided that every monotonic soft open cover of X has a finite subcover.
One can easily prove the following result.

Proposition 7. Every soft compact space is monotonic soft compact.
e next example elucidates that the converse of the above proposition is not always true.
In the following, we establish the main properties of a monotonic soft compact space which are similar to their counterparts on a soft compact space. Hence, (X, τ, E, ≺ ) is strongly p-soft regular ordered. □

Corollary 3. Every monotonic soft compact and strong p-soft
Definition 28 (see [28]). For a nonempty set X, a subcollection Λ of 2 X is said to have the finite intersection property (for short, FIP) if any finite subcollection of Λ has a nonempty intersection. Proof. Necessity: let Ω � (F j , E): j ∈ J be the collection of monotonic soft closed subsets of X which has the FIP.
Proof. Necessity: let Ω � (G j , E): j ∈ J be the collection of monotonic soft open subsets of (X, τ, E, ≺ ) which cover Y.
us, Y � ∪  It is clear that the limit points of a set (H, E) are a subset of the monotonic limit point of (H, E). e next example shows that the converse need not be true in general.
en, A l � a { } and A ml � X. Hence, A l is a proper subset of A ml . Theorem 9. Let (H, E) be a subset of (X, τ, E, ≺ ). en, the following results hold:  , E). erefore, X has at most n soft points of (H, E). is implies that (H, E) is finite, but this contradicts the infinity of (H, E). us, (H, E) has a monotonic soft limit point. en, e following is still an open problem.
□ Problem 1. Is the product of monotonic soft compact spaces a monotonic soft compact space?
Definition 31 (see [3]). A triple (X, τ, ≺ ) is said to be a topological ordered space if (X, ≺ ) is a partially ordered set and (X, τ) is a topological space.
Recall that (X, τ, ≺ ) is said to be monotonic compact if every monotonic open cover of X has a finite subcover. 6 Journal of Mathematics Proof. Let (G j , E): j ∈ J be a monotonic soft open cover of (X, τ, E, ≺ ). Without loss of generality, suppose that E � e 1 , e 2 . en, the collections G j (e 1 ): j ∈ J and G j (e 2 ): j ∈ J are monotonic open covers of (X, τ e 1 , ≺ ) and (X, τ e 2 , ≺ ), respectively. By hypothesis, there exist finitely two subsets M and N of J such that X � ∪ j∈M G j (e 1 ) and X � ∪ j∈N G j (e 1 ).
erefore, X � ∪ j∈M∪N (G j , E). us, (X, τ, E, ≺ ) is monotonic soft compact. To show that a finite condition of E is necessary, we give the following example. □ Example 5. Let a set of parameters be the set of natural numbers N, and let τ be a soft discrete topology on } is a partial order relation on X, then the collection Λ of all soft points of X is a monotonic soft open cover of X. Obviously, Λ has no finite subcover. erefore, (X, τ, N, ≺ ) is not monotonic soft compact. On the contrary, (X, τ e n , ≺ ) is soft compact for each n ∈ N.
Proof. Let H j (e): j ∈ J be a monotonic open cover of (X, τ e ). Since (X, τ, E, ≺ ) is extended, we choose all monotonic soft open sets (F j , E) such that F j (e) � H j (e) and F j (e i ) � X for all e i ≠ e. Obviously, (F j , E): j ∈ J is a monotonic soft open cover of (X, τ, E, ≺ ). By hypothesis, it follows that X � ∪ n j�1 (F j , E). us, X � ∪ n j�1 F j (e) � ∪ n j�1 H j (e). Hence, (X, τ e , ≺ ) is monotonic compact. Now, we give a condition which guarantees the converse of the above theorem holds. Proof. e proof follows from eorems 11 and 12.

Ordered Soft Compact Spaces and Applications on the Information System
is section presents a concept of ordered compact spaces and shows their relationships with monotonic compact and compact spaces. Also, it investigates their relationships with T i -ordered spaces (i � 2, 3, 4) and bicontinuous maps and shows that the product of ordered compact spaces need not be ordered compact. Finally, it gives an interesting application of ordered compact spaces on the information system.

Ordered Soft Compact Spaces
Definition 32. (X, τ, E, ≺ ) is said to be ordered soft compact if every soft open cover of X has a finitely monotonic subcover.
Proposition 11. Every ordered soft compact space is soft compact.

Corollary 5. Every ordered soft compact space is monotonic soft compact.
To see that the converse of the above two results need not be true, we give the next example.
} be a partial order relation on X. en, (X, τ, E, ≺ ) is compact because X is finite. Moreover, it is monotonic compact. On the contrary, the collection a } is an open cover of X. Since this collection has no finitely monotonic subcover, (X, τ, E, ≺ ) is not ordered compact.
e above example also shows that a finite topological space need not be ordered compact.

Definition 33.
e collection Λ � (F j , E): j ∈ J of S(X E ) is said to be minimal if every member of Λ covers some soft points of X which do not cover by any other members of Λ. In other words, removing any member (F j 0 , E) of Λ implies that Λ∖(F j 0 , E) is not a cover of X.

Definition 34.
e collection Λ � (F j , E): j ∈ J of P(X) is said to have the finite monotone property (FMP, in short) if all the minimal subcollections of Λ which covers X are monotonic.

Example 7.
Consider τ is the discrete topology on the set of real numbers R. en, the two collections } and x, y : x, y ∈ R are not minimal. On thecontrary, the two collections Proof. Necessity: it follows from Proposition 11. Sufficiency: let (G j , E): j ∈ J be a soft open cover of (X, τ, E, ≺ ). By compactness, we have X � ∪ n j�1 (G j , E). Now, (G j , E): j � 1, 2, . . . , n is a minimal collection covering X. Since (X, τ, E, ≺ ) has the FMP, (G j , E) is monotonic for each j � 1, 2, . . . , n. Hence, (X, τ, E, ≺ ) is ordered soft compact.

Theorem 13. Every ordered soft compact and p-soft
Proof.
e proof is similar to that of eorem 6. contradicts that Ω has the FIP for every monotonic soft set in this collection. Hence, Ω has a nonempty intersection. Sufficiency: let (G j , E): j ∈ J be a soft open cover of X. Suppose that (G j , E): j ∈ J has no finitely monotonic subcover. en, X\ ∪ n j�1 (G j , E) ≠ ∅ for each n ∈ N, where (G j , E) is a monotonic soft set for j � 1, 2, . . . , n. erefore, ∩ n j�1 (G j , E) c ≠ ∅ . is implies that (G j , E) c : j ∈ J is the collection of monotonic soft closed subsets of X which has the FIP.

Corollary 9. Every ordered soft compact and p-soft
us, ∩ j∈J (G j , E) c ≠ ∅ . us, X ≠ ∪ j∈J (G j , E), but this contradicts that Ω is a soft open cover of X. Hence, X is ordered soft compact.
e converse of the above proposition fails as illustrated in the following example.
this collection has no finitely monotonic subcover of Y. Hence, Y is not an ordered compact subset of (X, τ, E, ≺ ). We complete this section with the next example which shows that the product of ordered compact spaces need not be ordered compact. □ Example 9. Let τ and θ be the discrete topologies on X � a, b { } be the partial order relations on X and Y, respectively. One can check that (X, τ, ≺ 1 ) and (Y, θ, ≺ 2 ) are ordered compact spaces. Now, the product topology T on { } is the discrete topology. From Definition 12, the partial order relation ≺ on X × Y is given as follows: ≺ � △ ∪ ((a, 1), (a, 2)), ((a, { 1), (b, 1)), ((a, 1), (b, 2)), ((a, 2), (b, 2))}. It is clear that the collection Ω � (a, 1) Recall that (X, τ, ≺ ) is said to be ordered compact if every open cover of X has a finitely monotonic subcover.
Proof. Let H j (e): j ∈ J be an open cover of (X, τ e ). Since (X, τ, E, ≺ ) is extended, we choose all soft open sets (F j , E) such that F j (e) � H j (e) and F j (e i ) � X for all e i ≠ e. Obviously, (F j , E): j ∈ J is a soft open cover of (X, τ, E, ≺ ). By hypothesis, it follows that X � ∪ n j�1 (F j , E), where (F j , E) is monotonic for each j � 1, 2, . . . , n.

An Application of Ordered
Compactness on the Information System. In this part, we investigate an application of ordered compact spaces on the information system. It is well known that study compactness on the information system is meaningless because it is defined on a finite set so that this study is the first attempt of discussing a new type of compactness on the information system.
Definition 35 (see [31]). e information system is a pair of two nonempty sets (X, A) such that X is a finite set of objects and A is a finite set of attributes.
To start this part, consider Table 1 which represents a decision system.
It is well known that the equivalence classes form a soft basis for a soft topology on the universe set X such that every member of this basis is a soft clopen set.
We refer to a soft topology which was generated by the attributes a i and a i,j by τ a i and τ a i,j , respectively; and we refer to a soft topology which was generated by an attribute of decision d by τ d .
From Table 1, we generate a soft topology on X with respect to an attribute a 2 as follows.
First, we find the basis: Second, we find a soft topology on X from this soft basis: In a similar way, one can generate soft topologies on X with respect to the attributes a 1 , a 3 , and d.

Theorem 18. If (X, τ a i , E, ≺ ) is an ordered soft compact space, then its soft basis β a i is monotonic.
Proof. From the definition of a soft basis β a i , we find that the members of it are disjoint soft clopen sets such that X � ∪ j∈J (B j , E). Since X is finite, J must be finite, and since (X, τ a i , ≺ ) is an ordered soft compact space, all (B j , E) are monotonic soft sets. e converse of the above theorem is not always true as illustrated in the following example. en, a soft topology generated by other attributes, which are not superfluous, is ordered soft compact iff a soft topology generated by a decision attribute is ordered soft compact.

Conclusion
is research paper has studied the concepts of monotonic soft compact and ordered soft compact spaces using monotonic soft sets.
ese two concepts are considered as an extension of soft compact spaces. We have described them using the finite intersection property and have showed the relationship between them with the help of examples. Also, we have established some results related to soft ordered separation axioms and the finite product space. Furthermore, we have discussed preserving these concepts between the soft topological ordered space and its parametric soft topological ordered spaces. Finally, we give an interesting application of ordered soft compact spaces on the information system. In the upcoming works, we plan to investigate the following schemes: (i) Study the concepts of almost soft compact and mildly soft compact spaces on ordered setting (ii) Establish the concepts introduced in this work on some generalizations of soft topological ordered spaces such as soft bitopological ordered and supra soft topological spaces (iii) Carry out further studies concerning the applications of ordered soft compactness on the information system by making use of the application presented in [32,33]

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares that there are no conflicts of interest.