Abstract
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.
1. 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. The 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. Then, 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 -ordered and -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. Then, 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 Ayunoğlu and Ayun, and Zorlutuna et al. [9, 10]. Then, 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. Therefore, the authors of [13–17] carried out some corrective studies in this regard.
In 2019, Al-Shami et al. [18] defined topological ordered spaces on soft setting. They studied monotonic soft sets and then utilized them to present p-soft -ordered spaces. Also, they [19] presented soft -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 -ordered and strong soft -ordered spaces. Recently, the concept of supra soft topological ordered spaces has been explored and discussed in [21].
This 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.
2. 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.
2.1. Soft Set
Definition 1. (see [7]). A soft set over is a mapping from a parameter set to the power set of . It is denoted by .
Usually, we write a soft set as a set of ordered pairs. In other words, .
Sometimes, we use some symbols such as in place of and in place of .
Definition 2. over is said to be(i)a null soft set (resp. an absolute soft set) [22] if (resp. ) for each ; and it is denoted by (resp. ).(ii)soft point [23, 24] if there are and such that and for each . It is denoted by .(iii)finite (resp. countable) soft set [23] if is finite (resp. countable) for each . Otherwise, it is called infinite (resp. uncountable).
Definition 3. (see [25]). The relative complement of , denoted by , is a mapping defined by for each .
Definition 4. (see [26]). is a subset of if and for all .
Definition 5. (see [22]). The union of two soft sets and over , denoted by , is the soft set , where , and a mapping is defined as follows:
Definition 6. (see [25]). The intersection of soft sets and over , denoted by , is the soft set , where , and a mapping is defined by for all .
Definition 7. (see [9, 10]). Let and be two soft sets over and , respectively. Then, the Cartesian product of and is a soft set such that for each and .
Definition 8. (see [8, 27]). For a soft set over and , we say that(i) if for each , and if for some (ii) if for some , and if for each
Proposition 1. (see [10, 24]). For a soft mapping , we have the following results:(i) for each , and for each (ii)If is injective (resp. surjective), then (resp. )
Definition 9. (see [28]). A binary relation is called a partial order relation if it is reflexive, antisymmetric, and transitive. The pair is called a partially ordered set.
The relation on a nonempty set which is given by is called the equality relation and is denoted by .
Definition 10. (see [18]). is said to be a partially ordered soft set on if is a partially ordered set. For two soft points and in , we say that if .
Definition 11. (see [18]). Increasing operator and decreasing operator are two maps of into defined as follows: for each soft subset of ,(i), where is a mapping of into given by for some (ii), where is a mapping of into given by for some
Definition 12. Let and be two partially ordered soft sets. The product relation of and on is defined as follows: such that for every .
Definition 13. (see [18]). A soft subset of is said to be increasing (resp. decreasing) provided that (resp. ).
Theorem 1. (see [18]). The finite product of increasing (resp. decreasing) soft sets is increasing (resp. decreasing).
Definition 14. (see [18]). A soft map is said to be a subjective ordered embedding provided that if and only if .
Theorem 2. (see [18]). Let be a subjective ordered embedding soft mapping. Then, the image of each increasing (resp. decreasing) soft set is increasing (resp. decreasing).
Proposition 2. (see [18]).(i)The union of increasing (resp. decreasing) soft sets is increasing (resp. decreasing)(ii)The intersection of increasing (resp. decreasing) soft sets is increasing (resp. decreasing)
2.2. Soft Topological Space
Definition 15. (see [8]). The family of soft sets over under a fixed parameter set is said to be a soft topology on if it contains and and is closed under a finite intersection and an arbitrary union.
The triple is said to be a soft topological space. Every member of is called soft open, and its relative complement is called soft closed.
Definition 16. (see [8]). A soft subset of is called soft neighborhood of if there exists a soft open set such that .
Definition 17. (see [10, 24]). A soft mapping is said to be(i)soft continuous if the inverse image of each soft open set is soft open(ii)soft open (resp. soft closed) if the image of each soft open (resp. soft closed) set is soft open (resp. soft closed)(iii)soft bicontinuous if it is soft continuous and soft open(iv)soft homeomorphism if it is bijective soft bicontinuous
Theorem 3. (see [9]). Let and be two soft topological spaces. Let and . Then, the family of all arbitrary union of elements of is a soft topology on .
Definition 18. (see [29]). Let be a soft topological space. Then,is called an extended soft topology on .
Many properties of extended soft topologies which help us to show the relationships between soft topology and its parametric topologies were studied in [30].
2.3. Soft Topological Ordered Space
Definition 19. (see [18]). A quadrable system is said to be a soft topological ordered space if is a soft topological space and is a partially ordered set.
Definition 20. (see [18]). A soft subset of is said to be increasing (resp. decreasing) soft neighborhood of if is increasing (resp. decreasing) and a soft neighborhood of .
Proposition 3. (see [18]). In , we find that, for each , the family with a partial order relation forms an ordered topology on .
is said to be a parametric topological ordered space.
Definition 21. (see [18]). Let . Then, is called a soft ordered subspace of if is a soft subspace of and .
Lemma 1. (see [18]). If is an increasing (resp. a decreasing) soft subset of , then is an increasing (resp. a decreasing) soft subset of a soft ordered subspace .
Definition 22. (see [18]). The product of a finite family of soft topological ordered spaces is a soft topological ordered space , where , is the product soft topology on , , and such that for every .
Definition 23. (see [18]). A soft ordered subspace of is called soft compatibly ordered if for each increasing (resp. decreasing) soft closed subset of , there exists an increasing (resp. a decreasing) soft closed subset of such that .
Definition 24. (see [18]). is said to be(i)p-soft -ordered if for every distinct points in , there exist disjoint soft neighborhoods and of and , respectively, such that is increasing and is decreasing(ii)lower (upper) p-soft regularly ordered if for each decreasing (increasing) soft closed set and such that , there exist disjoint soft neighbourhoods of and of such that is decreasing (increasing) and is increasing (decreasing)(iii)p-soft regularly ordered if it is both lower p-soft regularly ordered and upper p-soft regularly ordered(iv)p-soft -ordered if it is both lower p-soft -ordered and upper p-soft -orderedIf the phrase “soft neighborhoods” is replaced by “soft open sets,” then the above soft axioms are called strong p-soft regularly ordered and strong p-soft -ordered spaces, .
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.
Definition 25. A subset of is called a monotonic soft set if is increasing or decreasing. In other words, or .
The following example shows that the union and intersection of monotonic soft sets are not always monotonic.
Example 1. Let be a partial order relation on , and let be a set of parameters. We define the following three subsets of as follows:Since , , and , , , and are monotonic soft sets. Now, and . It is clear that and . Hence, and are not monotonic soft sets.
The following result demonstrates under what conditions the union (intersection) of monotonic soft sets is monotonic.
Proposition 4. Let and be two subsets of and , respectively. If and are both increasing (decreasing), then the union and intersection of and are monotonic soft sets.
Proof. It follows immediately from Proposition 2.
Corollary 1. Let and be two monotonic subsets of and , respectively. Then, we have the following two results:(i) and are monotonic or and are monotonic(ii) and are monotonic or and are monotonic
Proposition 5. Let and be two soft subsets of and , respectively. We have the following two results:(i)(ii)
Proof. We only prove (i), and one can prove (ii) similarly.
: there exists such that .(i) and : there exist and such that and (ii): there exist such that and : there exist such that (iii)
The next example illustrates that the product of two monotonic soft sets need not be a monotonic soft set.
Example 2. Let and be the partial order relations on and , respectively. From Definition 12, the partial order relation on is given as follows: . Take as a set of parameters. Now, is a monotonic subset of because ; and is a monotonic subset of because . On the contrary, their product , is not a monotonic subset of because and , .
The following result demonstrates under what conditions the product of two monotonic soft sets is a monotonic soft set.
Proposition 6. Let and be two subsets of and , respectively, such that and are both increasing (decreasing). Then, and are monotonic soft sets iff is a monotonic soft set.
Proof. It follows immediately from Proposition 5.
In view of Theorem 2, we obtain the following result.
Theorem 4. If is an order embedding map, then(i)The image of each monotonic soft set is monotonic(ii)The inverse image of each monotonic soft set is monotonic
4. 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 -ordered spaces . Finally, we investigate some results that associated the concept of monotonic compact spaces with monotonic limit points and some types of monotonic maps.
Definition 26. The collection of soft open subsets of is called a monotonic soft open cover for provided that and all are monotonic.
Definition 27. is said to be monotonic soft compact provided that every monotonic soft open cover of has a finite subcover.
One can easily prove the following result.
Proposition 7. Every soft compact space is monotonic soft compact.
The next example elucidates that the converse of the above proposition is not always true.
Example 3. Let such that be the particular point soft topology on the set of natural numbers , be a set of parameters, and is smaller than be a partial order relation on . Obviously, is not soft compact. On the contrary, the only increasing open subsets of are and . The decreasing open subsets of are given in the form for each . Hence, is monotonic soft compact.
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.
Proposition 8. Every monotonic soft closed subset of a monotonic soft compact space is monotonic soft compact.
Proof. Let be a monotonic soft open cover of a monotonic soft closed subset of . Since is monotonic soft open, is a monotonic soft open cover of . Therefore, . Thus, . Hence, is monotonic soft compact.
Corollary 2. The intersection of monotonic soft closed and monotonic soft compact sets is monotonic soft compact.
Theorem 5. Let be a monotonic soft compact subset of a strong p-soft -ordered space . If , then there exist disjoint monotonic soft open sets and containing and , respectively.
Proof. Let be a monotonic soft closed set such that and . Without loss of generality, suppose that is increasing. Then, . Since is strong p-soft -ordered, there are an increasing soft open set containing and a decreasing soft open set containing such that and are disjoint. Therefore, forms an increasing soft open cover of . Thus, . Obviously, the two disjoint soft open sets and are increasing and decreasing, respectively. Hence, the proof is completed.
Theorem 6. Every monotonic soft compact and strong p-soft -ordered space is strongly p-soft regular ordered.
Proof. Let be a monotonic soft closed subset of such that . Since is monotonic soft compact, is monotonic soft compact. Since is strong p-soft -ordered, it follows from Theorem 5 that there are two disjoint monotonic soft open sets and containing and , respectively. Hence, is strongly p-soft regular ordered.
Corollary 3. Every monotonic soft compact and strong p-soft -ordered space is strong p-soft -ordered.
Definition 28. (see [28]). For a nonempty set , a subcollection of is said to have the finite intersection property (for short, FIP) if any finite subcollection of has a nonempty intersection.
Theorem 7. is a monotonic soft compact space iff every collection of monotonic soft closed subsets of , satisfying the FIP, has a nonempty soft intersection.
Proof. Necessity: let be the collection of monotonic soft closed subsets of which has the FIP. Suppose that . Then, . Since is monotonic soft compact, . Therefore, , but this contradicts that has the FIP. Hence, has a nonempty soft intersection.
Sufficiency: let be a monotonic soft open cover of . Suppose that has no finitely monotonic subcover. Then, for each . Therefore, . This implies that is the collection of monotonic soft closed subsets of which has the FIP. By hypothesis, . Thus, , but this contradicts that is a monotonic soft open cover of . Hence, is monotonic soft compact.
Theorem 8. The subspace of is monotonic soft compact iff is a monotonic soft compact set in .
Proof. Necessity: let be the collection of monotonic soft open subsets of which cover . Then, is the collection of monotonic soft open subsets of which cover . By hypothesis, we have . Thus, . Hence, is a monotonic soft compact set.
Sufficiency: let be the collection of monotonic soft open subsets of which cover . Since for some , is a monotonic soft open cover of . By hypothesis, we have . Thus, . Hence, is monotonic soft compact.
Definition 29. Let be a soft subset of and . A soft point is said to be a monotonic soft limit point of if for every monotonic soft open set containing .
The soft set of all monotonic soft limit points of is denoted by .
It is clear that the limit points of a set are a subset of the monotonic limit point of . The next example shows that the converse need not be true in general.
Example 4. Consider is a topology on , and let be a partial order relation on . Let . Then, and . Hence, is a proper subset of .
Theorem 9. Let be a subset of . Then, the following results hold:(i) is monotonic soft closed iff (ii) is monotonic soft closed
Proof. We only prove (i), and one can prove (ii) similarly.
Necessity: assume that is a monotonic soft closed set and . Then, . Since is monotonic soft open and , . Therefore, .
Sufficiency: let and . Then, . Therefore, there is a monotonic soft open set such that . Since , . Now, . Therefore, . Thus, is a monotonic soft open set. Hence, is monotonic soft closed.
Corollary 4. If is a soft subset of a monotonic soft compact space , then is monotonic soft compact.
Theorem 10. Every infinite soft subset of a monotonic soft compact space has a monotonic soft limit point.
Proof. Suppose that is an infinite soft subset of a monotonic soft compact space . Suppose that does not have a monotonic soft limit point. Then, for each , we have a monotonic soft open set containing such that . Now, the collection forms a monotonic soft open cover of . Since is monotonic soft compact, . Therefore, has at most soft points of . This implies that is finite, but this contradicts the infinity of . Thus, has a monotonic soft limit point.
Definition 30. A soft map is said to be(i)monotonic soft continuous if the preimage of every monotonic soft open set is a monotonic soft open set(ii)monotonic soft open (resp. monotonic soft closed) if the image of every monotonic soft open (resp. monotonic soft closed) set is a monotonic soft open (resp. monotonic soft closed) set(iii)monotonic soft homeomorphism if it is bijective, monotonic soft continuous, and monotonic soft open
Proposition 9. The property of being a monotonic soft compact set is preserved under a monotonic soft continuous map.
Proof. Let be a monotonic soft continuous map, and let be a monotonic soft compact subset of . Suppose that is a monotonic soft open cover of . Then, . Since is monotonic soft continuous, is a monotonic soft open set for all . Since is monotonic soft compact, . Hence, is monotonic soft compact.
The 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 is said to be a topological ordered space if is a partially ordered set and is a topological space.
Recall that is said to be monotonic compact if every monotonic open cover of has a finite subcover.
Theorem 11. Let be monotonic compact for each . Then, is a monotonic soft compact space if is finite.
Proof. Let be a monotonic soft open cover of . Without loss of generality, suppose that . Then, the collections and are monotonic open covers of and , respectively. By hypothesis, there exist finitely two subsets and of such that and . Therefore, . Thus, is monotonic soft compact.
To show that a finite condition of is necessary, we give the following example.
Example 5. Let a set of parameters be the set of natural numbers , and let be a soft discrete topology on . If is a partial order relation on , then the collection of all soft points of is a monotonic soft open cover of . Obviously, has no finite subcover. Therefore, is not monotonic soft compact. On the contrary, is soft compact for each .
Theorem 12. Let be an extended soft topological space. If is a monotonic soft compact space, then is monotonic compact for each .
Proof. Let be a monotonic open cover of . Since is extended, we choose all monotonic soft open sets such that and for all . Obviously, is a monotonic soft open cover of . By hypothesis, it follows that . Thus, . Hence, is monotonic compact.
Now, we give a condition which guarantees the converse of the above theorem holds.
Proposition 10. Let be an extended soft topological space such that is finite. Then, is a monotonic soft compact space iff is monotonic compact for some .
Proof. The proof follows from Theorems 11 and 12.
5. Ordered Soft Compact Spaces and Applications on the Information System
This section presents a concept of ordered compact spaces and shows their relationships with monotonic compact and compact spaces. Also, it investigates their relationships with -ordered spaces 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.
5.1. Ordered Soft Compact Spaces
Definition 32. is said to be ordered soft compact if every soft open cover of has a finitely monotonic subcover.
Proposition 11. Every ordered soft compact space is soft compact.
Proof. Straightforward.
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.
Example 6. Consider is the discrete topology on , and let be a partial order relation on . Then, is compact because is finite. Moreover, it is monotonic compact. On the contrary, the collection is an open cover of . Since this collection has no finitely monotonic subcover, is not ordered compact.
The above example also shows that a finite topological space need not be ordered compact.
Definition 33. The collection of is said to be minimal if every member of covers some soft points of which do not cover by any other members of . In other words, removing any member of implies that is not a cover of .
Definition 34. The collection of is said to have the finite monotone property (FMP, in short) if all the minimal subcollections of which covers are monotonic.
Example 7. Consider is the discrete topology on the set of real numbers . Then, the two collections and are not minimal. On thecontrary, the two collections and are minimal.
Proposition 12. If every collection of the soft open cover of satisfies the FMP, then is ordered soft compact iff it is soft compact.
Proof. Necessity: it follows from Proposition 11.
Sufficiency: let be a soft open cover of . By compactness, we have . Now, is a minimal collection covering . Since has the FMP, is monotonic for each . Hence, is ordered soft compact.
Proposition 13. Every soft closed subset of an ordered soft compact space is ordered soft compact.
Proof. Let be a soft open cover of a soft closed subset of . Then, is soft open. Therefore, is a soft open cover of . By hypothesis, is ordered soft compact; then, , where is a monotonic soft open set for each . Thus, . Hence, is ordered soft compact.
Corollary 6. The intersection of soft closed and ordered soft compact sets is ordered soft compact.
Corollary 7. Let be a soft closed subset of an ordered soft compact space such that is not monotonic. Then, every monotonic cover of is also a monotonic cover of .
Lemma 2. Let be a monotonic and ordered soft compact subset of a p-soft -ordered space . If , then there are two monotonic soft neighbourhoods and of and , respectively, such that .
Proof. Let be a monotonic ordered soft compact set such that and . Without loss of generality, suppose that is increasing. Then, . Since is p-soft -ordered, there are an increasing soft neighbourhood of and a decreasing soft neighbourhood of such that and are disjoint. Therefore, there is a soft open set containing such that . Now, is a soft open cover of . By hypothesis, , where is a monotonic soft open set for each . Thus, . Obviously, is a decreasing neighbourhood of , and is an increasing neighbourhood of such that and are disjoint. Hence, the proof is completed.
Theorem 13. Every ordered soft compact and p-soft -ordered space is p-soft regularly ordered.
Proof. The proof is similar to that of Theorem 6.
Corollary 9. Every ordered soft compact and p-soft -ordered space is p-soft -ordered.
Theorem 14. is an ordered soft compact space iff every collection of soft closed subsets of , satisfying the FIP for the monotonic soft sets in this collection, has a nonempty intersection.
Proof. Necessity: let be the collection of soft closed subsets which has the FIP for the monotonic soft sets in this collection. Suppose that . Then, . Since is ordered soft compact, , where is a monotonic soft set for each . Therefore, , but this contradicts that has the FIP for every monotonic soft set in this collection. Hence, has a nonempty intersection.
Sufficiency: let be a soft open cover of . Suppose that has no finitely monotonic subcover. Then, for each , where is a monotonic soft set for . Therefore, . This implies that is the collection of monotonic soft closed subsets of which has the FIP. Thus, . Thus, , but this contradicts that is a soft open cover of . Hence, is ordered soft compact.
Proposition 14. If is an ordered soft compact set in , then a subspace of is ordered soft compact.
Proof. Let be the collection of soft open subsets of which covers . Since for some , is a soft open cover of in . By hypothesis, we have , where is a monotonic soft set for . Thus, , where is a monotonic soft set for in . Hence, is ordered soft compact.
The converse of the above proposition fails as illustrated in the following example.
Example 8. Let be the same as in Example 6. If , then is the discrete topology on , and is a partial order relation on . It is clear that is an ordered compact space. On the contrary, the collection is an open cover of in . Since and , this collection has no finitely monotonic subcover of . Hence, is not an ordered compact subset of .
Theorem 15. Every infinite subset of an ordered soft compact space has a soft limit point.
Proof. The proof is similar to that of Theorem 10.
Theorem 16. The property of being an ordered soft compact set is preserved under a soft bicontinuous surjective and soft ordered embedding map.
Proof. Let be a soft bicontinuous map, and let be an ordered soft compact subset of . Suppose that is a soft open cover of . Then, . Since is soft continuous, is a soft open set for all . By hypothesis, is ordered soft compact; then, , where is a monotonic soft set for each . Since is soft open and soft ordered embedding, is a monotonic soft open set for each . Now, . Since is surjective, for each . Thus, where is a monotonic soft open set for each . Hence, is an ordered soft compact set.
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 and be the discrete topologies on and , respectively. Let and be the partial order relations on and , respectively. One can check that and are ordered compact spaces. Now, the product topology on is the discrete topology. From Definition 12, the partial order relation on is given as follows: . It is clear that the collection forms an open cover of . Since and , is not a monotonic subset of . Since is minimal, is not an ordered compact space.
Recall that is said to be ordered compact if every open cover of has a finitely monotonic subcover.
Theorem 17. Let be an extended soft topological space. If is an ordered soft compact space, then is ordered compact for each .
Proof. Let be an open cover of . Since is extended, we choose all soft open sets such that and for all . Obviously, is a soft open cover of . By hypothesis, it follows that , where is monotonic for each . Thus, such that is monotonic for each . Hence, is ordered compact.
5.2. 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]). The information system is a pair of two nonempty sets such that is a finite set of objects and 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 such that every member of this basis is a soft clopen set.
We refer to a soft topology which was generated by the attributes and by and , respectively; and we refer to a soft topology which was generated by an attribute of decision by .
From Table 1, we generate a soft topology on with respect to an attribute as follows.
First, we find the basis: .
Second, we find a soft topology on from this soft basis: , , , , , .
In a similar way, one can generate soft topologies on with respect to the attributes , , and .
Theorem 18. If is an ordered soft compact space, then its soft basis is monotonic.
Proof. From the definition of a soft basis , we find that the members of it are disjoint soft clopen sets such that . Since is finite, must be finite, and since is an ordered soft compact space, all are monotonic soft sets.
The converse of the above theorem is not always true as illustrated in the following example.
Example 10. From Table 1, a soft basis with respect to an attribute is , . A soft topology on from this soft basis is , , , , , , , .
Let be a partial order relation on . Then, the basis is monotonic. On the contrary, the collection is a soft open cover of . Since this a soft open cover does not have a finitely monotonic subcover, is not ordered soft compact.
Theorem 19. Let be superfluous attributes. Then, 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.
Proof. Without loss of generality, suppose that is a superfluous attribute. Then, a soft topology generated by all attributes such that is identical with a soft topology generated by a decision attribute. Hence, the desired result is proved.
In the rest of this part, we study the importance of ordered soft compactness to expect the missing values of the table of the information system. To this end, we present an algorithm for expecting the missing values on the information system, and then we provide two illustrative examples.
An algorithm for expecting the missing values on the information system is given.(1)Determine the objects (which are soft sets here) with a missing value with respect to the given attribute. Say, these objects are , where .(2)Classify these objects in terms of increasing and decreasing with respect to the given partial order relation in the following three cases:(i)Neither increasing nor decreasing(ii)Increasing, but not decreasing, or decreasing, but not increasing(iii)Increasing and decreasing(3)Write the soft basis which generated a soft topology , where .(4)If there is a nonmonotonic member of , then add the sufficient objects with the missing value to make it monotonic. Otherwise, go to the next step.(5)Examine the objects with the missing value according to the order of them in Step 2.(6)Write the expected values for each missing value object.The following two examples are the application of carrying out this algorithm.
Example 11. Consider is an ordered compact space, where is a topology generated by an attribute , which is illustrated in Table 2, on the universe set , and let be a partial order relation on .
According to Table 2, the basis of a soft topology on is , . It is clear that the only object with the missing value is . This object is neither increasing nor decreasing. Now, the ordered soft compactness of implies that a basis must be monotonic. However, a member of is nonmonotonic. Thus, must belong to the member . In other words, the expected value of is 3. Hence, Table 2 shall take the following form.
Example 12. Consider is an ordered soft compact space, where is a soft topology generated by an attribute , which is illustrated in Table 3, on the universe set , and let , be a partial order relation on .
According to Table 3, the basis of a soft topology on is . Since is ordered soft compact, a basis must be monotonic, but a member of is neither increasing nor decreasing so that we add to . Thus, the expected value of is 3. Hence, we have a new basis , on .
Now, we process the remaining two missing values separately. Since is a decreasing singleton, but not increasing, and is an increasing and decreasing singleton, we choose secondly according to item 2 of the algorithm given above. It is clear that if we add to any decreasing member of , we obtain a monotonic basis so that we have the following three cases for the new basis :(i)(ii)(iii)Hence, the expected values of are 1, 2, or 3.
Finally, is a decreasing and an increasing singleton; it can be added to any member of so that we have the following cases for the new basis:(i)(ii)(iii)(iv)(v)(vi)(vii)(viii)(ix)(x)(xi)(xii)(xiii)(xiv)(xv)Hence, the expected values of are 1, 2, 3, or 5, or any value different than them.
6. Conclusion
This research paper has studied the concepts of monotonic soft compact and ordered soft compact spaces using monotonic soft sets. These 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
The author declares that there are no conflicts of interest.