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Categorical Properties of Soft Sets
The present study investigates some novel categorical properties of soft sets. By combining categorical theory with soft set theory, a categorical framework of soft set theory is established. It is proved that the category SFun of soft sets and soft functions has equalizers, finite products, pullbacks, and exponential properties. It is worth mentioning that we find that SFun is both a topological construct and Cartesian closed. The category SRel of soft sets and -soft set relations is also characterized, which shows the existence of the zero objects, biproducts, additive identities, injective objects, projective objects, injective hulls, and projective covers. Finally, by constructing proper adjoint situations, some intrinsic connections between SFun and SRel are established.
It is well known that many traditional mathematical tools such as fuzzy set theory, probability theory, rough set theory, and interval mathematic theory have their own limitations in dealing with some uncertain problems caused by the incompatibility of various parameter tools. To overcome the difficulties mentioned above, Molodtsov  initiated soft set theory by introducing enough compatible parameters. In the context of soft set, researchers can choose freely the form of parameters to simplify the decision-making process, which often makes the process more efficient under the absence of partial information. Consequently, Ali et al.  further introduced some new operations in soft set theory. Recently, soft set theory has opened up keen insights and has a rich potential for application in many different fields such as ontology , data analysis [4, 5], forecasting , simulation , decision making [8–11], medical science , rule mining , algebraic systems [14–20], optimization , and textures classification . However, being originated from relatively simple information models, classical soft set theory may not be suitable for those complex information models. In order to solve practical problems better by employing soft set theory, it is important to allure capable pure mathematicians to participate in the study of soft set theory. On the other hand, category theory is not only a basic tool for characterizing all kinds of mathematical structures, but also a tie which can connect easily the fields of mathematics and theoretical computer science (see [23–28]). Many researchers (see ) even argue that it is category theory, rather than set theory, that provides the proper setting for the study of pure mathematics. Based on the above analysis, a natural question is whether we can research by combining soft set theory with category theory. The fact is that there exists some categorical concepts, such as product, in soft set theory. Moreover, category theory has been successfully applied to fuzzy set theory [30, 31] and rough set theory [32, 33]. In 2007, Aktaş and Çağman  showed that both a fuzzy set and a rough set can be regarded as a soft set, which makes it possible to investigate soft set theory and category theory in a common setting. Inspired by this, recently, Zahiri  introduced a category whose objects are soft sets. Sardar and Gupta  defined another soft category which is a parameterized family of subcategories of a category. Varol et al.  defined a new category of soft sets and soft mapping. These studies have presented a preliminary, but potentially interesting, research direction. However, some basic problems still need further investigation. Based on these analyses, we further study the categorical framework of soft set theory in the present paper.
The main contributions of the paper have 3-fold. First, we show that the category SFun of soft sets and soft functions is Cartesian closed. On the one hand, because of the consistency of expression function between Cartesian closed category and -calculation with types, many researchers have been devoted to establishing all kinds of Cartesian closed categories in the universe theory for denotational semantics of computer programming language. On the other hand, soft set theory has been widely applied to many fields. Based on this, we further study the category SFun of soft sets and soft functions and prove that it is Cartesian closed. Second, we give a new characterization on soft set relations by employing category theory. There is no doubt that soft set relations play a significant role in the study of soft set theory and they can not only characterize the theoretical relations of two soft sets but also enrich the soft set theory. Presently, researches on soft set relations have received widespread attention and have made great progress (see [37–42]). Meanwhile, it is worth noting that category of binary relations has been widely applied to mathematics and computer science [43, 44]. Inspired by this, we make a further discussion on the category SRel of soft sets and -soft set relations. Third, we construct a concrete adjoint situation between the category SFun and SRel and characterize its basic relationships.
The remaining parts of the paper are arranged as follows. Section 2 shows some preliminaries. We present in Section 3 the concept of soft functions and discuss the fundamental properties of the category SFun. In Section 4, the characterizations of the category SRel are investigated. Section 5 focuses on studying the intrinsic connections between SFun and SRel.
In this section, we recall some elementary notions and facts related to soft set theory , category theory (see [23–27]) which will be often used in this paper. In what follows, we denote by an initial universe of objects and by the set of parameters that relate to objects in . presents the power set of . , , , and are the subsets of .
Definition 1 (see ). A pair (, ) is called a soft set over , where is a function given by .
In other words, a soft set over is a parameterized family of subsets of . For any parameter , may be considered as the set of -approximate elements of the soft set .
Proposition 2 (see ). If a category has finite products and equalizers, then has pullbacks.
Definition 3 (see ). Let be an object in a category . One calls initial if for each object there is exactly one morphism from to ; one calls terminal if for each object there is exactly one morphism from to ; and one calls a zero object if it is both initial and terminal.
For objects in a category with zero object , we use for the unique morphism .
Definition 4 (see ). A category with zero has biproducts if for each family of objects there is an object , together with families of morphisms and such that (i)the morphisms are a coproduct of the family ;(ii)the morphisms are a product of the family ;(iii) for each ; here is the identity map if and the zero map if .
Example 5 (see ). In category Rel of sets and the relations between them, for a family of sets , let be their disjoint union and define relations from to and from to by setting and . Then the disjoint union with morphisms and is a biproduct of the family .
Definition 6 (see ). A semiadditive category is a category where each homset is equipped with the structure of a commutative monoid with operation + such that, for any , and ,
Definition 7 (see ). An involution on a category is a contravariant functor from to itself of period two.
Definition 8 (see ). Let be a -object. Then is injective if, for every monic and each , there is an with : (2) The map is called an injective hull of if is monic, is injective, and for any we have being monic which implies that is monic.
Definition 9 (see ). For categories and and functors and , one says is an adjoint situation if is left adjoint to and is right adjoint to . This implies that, for objects and , there is a natural isomorphism between the homsets .
Definition 10. One calls that a category has exponential properties if it has finite products and for each of the -objects , there exists a -object and a -morphism such that, for each -object and -morphism , there exists a unique -morphism with . That is, the diagram
Definition 11 (see ). A category is called Cartesian closed if it has equalizers, finite products, terminal objects, and exponential properties.
For the other standard terminology of category theory, see [24, 26].
3. The Category SFun of Soft Sets and Soft Functions
The properties of the category SFun will be investigated in this section. Particularly, we will prove that SFun is a topological construct and Cartesian closed.
Definition 12. Let and be two soft sets over . Then one says that the mapping is a soft function from to if it satisfies for each .
Example 13. Let be the set of candidate dresses and the set of parameters, where stands for expensive, beautiful, elegant, and classical, respectively. Let , , , , , and . It is easy to check that and are two soft sets over . Define a function by . By routine calculations, we can prove that and . By Definition 12, is a soft function from to .
Remark 14. The concept of soft functions is different from soft set functions defined in .
Let SFun denote the category of all soft sets over and soft functions. We next discuss the properties of the category SFun.
Lemma 15. has equalizers (4)
Proof. Suppose that and are two SFun-objects over ; and are two SFun-morphisms from to . Define , , an embedding, and . From the assumption, we can easily know that is a SFun-object, , and for each . Thus is a SFun-morphism. We next show that is the equalizer of and . Assume that is a SFun-object and is a SFun-morphism from to satisfying . Define a mapping and . In what follows we focus on showing that is a SFun-morphism from to and . Firstly, by , we can infer that for each , which means that . Hence is well defined. Secondly, according to , and being a SFun-morphism, we have where . Therefore, is a SFun-morphism. At last, from the assumption, we know that and is unique. In conclusion, is the equalizer of and .
Lemma 16. has finite products (6)
Proof. Firstly, let and be two SFun-objects. Define three mappings whence, for each , It follows that is a SFun-morphism. By the same argument, is also a SFun-morphism. Secondly, for each SFun-object , suppose that and are SFun-morphisms from to and , respectively. Then and for every . Further, define a mapping Then we can infer that, for each , which yields that is a SFun-morphism. At last, for every , one obtains Therefore, . Analogously, . Apparently, is unique. In conclusion, is a finite product of and .
Theorem 17. SFun has pullbacks.
Proof. By Proposition 2, it is a direct consequence of Lemmas 15 and 16.
Lemma 18. has terminal objects.
Proof. Define a mapping Trivially, is a SFun-object. For every SFun-object , define a mapping Then for each , it holds that which implies that is a SFun-morphism from to . It is easy to know that is unique. By Definition 3, is a terminal object of SFun.
Proposition 19. has initial objects.
Proof. The proof runs parallel to that of Lemma 18.
According to Definition 3, we can easily obtain the following proposition.
Proposition 20. has zero objects.
Lemma 21. has exponential properties.
Proof. Assume that and are two SFun-objects over ; . For all , define
Then . In fact, choose and define a mapping by such that . whence , which meas that . That is, .
From the definitions, we can easily know that is a SFun-object. Define the evaluation mapping as follows: Then for all . It is immediate that which implies that is a SFun-morphism. Furthermore, we show that has the couniversal property. Assume that is a SFun-object such that for every and is a SFun-morphism. It remains to prove that there exists a unique SFun-morphism such that . Firstly, for every , define Since is a SFun-morphism, one has Consequently, which implies that . Further, according to the assumption, . Hence and is a SFun-morphism from to . Secondly, for each , it holds that whence . Namely, the diagram (23)
is commutative. Furthermore, suppose that is a SFun-morphism satisfying . Then for every and , On the other hand, we have Thus . Since and are arbitrary, . This completes the proof.
Now we are ready to present two of our main results as follows.
Theorem 22. is Cartesian closed.
Proof. Lemmas 15, 16, 18, and 21 prove the claim.
Theorem 23. is a topological construct.
Proof. Let be a family of SFun-objects indexed by a class and a family of mappings. Define a soft set over as follows:
Then b(SFun). It suffices to show that is the unique SFun initial lift of . Next, we complete the proof by the following two steps.
Step 1. We show that is a SFun initial lift of . Firstly, we claim that is a family of SFun-morphisms for every . By the assumption, for each and , one yields whence is a family of SFun-morphisms. Furthermore, suppose that b(SFun), is a mapping such that for every , and is a family of SFun-morphisms. Then, we can infer that for all and . It follows that Therefore, is a SFun-morphism from to . By definition, we can know that is a SFun initial lift of .
Step 2. We show the uniqueness of the initial lift. If is also a SFun initial lift of which is different from , then is a family of SFun-morphisms. It is immediate that for each and . Consequently, . That is, . On the other hand, for the SFun-object and identity mapping , since is a SFun initial lift of , we have , is a family of SFun-morphisms for all , and is also a SFun-morphism. Therefore, for each , which means that . To sum up, . Based on Steps 1 and 2, SFun is a topological construct.
4. The Category SRel of Soft Sets and -Soft Set Relations
The main aim of this section is to investigate the properties of the category SRel. We will begin with the analysis of the existence of the zero object, biproduct, and additive identity of SRel. Then the injective object, projective object, injective hull, and projective cover of SRel will be studied.
Definition 24 (see ). Let and be two soft sets over . Then the Cartesian product of and is defined as , where and for all ; that is, .
Definition 25 (see ). Let and be two soft sets over . Then a relation from to is a soft subset of .
In other words, a relation from to is of the form , where and , for every ; here has been defined in Definition 24. Any subset of is called a relation on .
From Definition 25, we can see that the condition for soft set relation between two soft sets is very weak but just the weak conditions of the soft set relation make those many elements satisfying the above definition in fact unrelated in actual problems. It can be illustrated clearly by the following example.
Example 26. Consider the soft set which describes “the cost of the mobile phones” and the soft set which describes the “attractiveness of mobile phones.” Assume that is the universe consisting of six mobile phones, and the parameter sets is given by and , respectively, where stands for “very cheap,” “costly,” “very costly,” “beautiful,” and “accessible,” respectively. Let , , , , , and . Let . The relation from to is given by , where for every . By Definition 25, .
In the above example, , . It is obvious that there is no relation between them. However, , whence Definition 25 cannot describe precisely the relation between soft sets. To overcome this limitation, we strengthen the concept of soft set relations by defining a new soft set relation.
Definition 27. Let and be two soft sets over . Then a -soft set relation from to is a subset of defined as
The definition can be illustrated by diagrams of the form(30)
Example 28. Let and be the soft sets defined in Example 26. By Definition 27, we have .
Definition 29. Let be a -soft set relation from to and a -soft set relation from to . Then the composition of and , denoted by , is defined as follows:
Definition 30. Let be a soft set over . The identity -soft set relation on is defined as .
Proposition 31. Let be a -soft set relation from to , a -soft set relation from to , and an identity -soft set relation on . Then and .
Remark 32. From the aforementioned definitions and propositions, we can construct a category, denoted by SRel, whose objects are all soft sets and morphisms are all -soft set relations.
Proposition 33. The category of soft sets and soft functions is the subcategory of .
Proposition 34. The empty set (with the empty function into ) is a zero object in .
Proof. For each SRel-object , there exist unique morphisms
The inequalities are satisfied by default, whence the sets and each contain exactly one morphism.
Proposition 35. has biproducts.
Proof. Let be a family of SRel-objects and the disjoint union of . Define a mapping as . Further, define relations from to and from to as follows:
We next show that with morphisms and is a biproduct of the family by the following four steps.
Step 1. We first prove that and are SRel-morphisms. In fact, take an element in ; since for each and , we have , which means that is a -soft set relation from to . That is, is a SRel-morphism from to for each . Analogously, we can prove that is a SRel-morphism from to for every .
Step 2. We show that are the morphisms for a coproduct. Suppose that is a SRel-object and is a family of SRel-morphisms. Define a relation from to by if and only if for each . Firstly, we claim that is a SRel-morphism from to . If for each and , since is a family of SRel-morphisms for each , we have for every and . On the other hand, , which implies that . By Definition 27, we have , whence is a SRel-morphism from to . Secondly, we prove that for all . Let and ; then by Definition 29, is equivalent to and for some . According to assumption, if and only if , whence . At last, the uniqueness is obvious. In conclusion, are the morphisms for a coproduct.
Step 3. We further show that are morphisms for a product. Assume that is SRel-object and is a family of SRel-morphisms. Define a relation from to by setting if and only if . Similar to Step 2, we can infer that is a unique morphism from to in SRel with . Thus are morphisms for a product.
Step 4. Finally, a calculation shows that is the identical relation on if and the empty relation from to if . Therefore, .
From the above discussion, we know that SRel has biproducts by Definition 4.
Any category with biproducts carries a unique semiadditive structure that can be defined via biproducts . Next we briefly describe some properties of SRel.
Proposition 36. Let and be two -morphisms from to . Then the semiadditive structure on homesets in is given by taking to be the union of -soft set relations . In this case, the empty -soft set relation serves as the additive identity.
Proof. Let and be two SRel-morphisms from to . Firstly, we show that is a SRel-morphism from to . In fact, let , and ; then or . In the first case, is a SRel-morphism given by . And in the second case, is a SRel-morphism given by , whence is a morphism in SRel. Secondly, we can easily verify that gives a commutative monoid structure on with the empty -soft set relation as identity, and composition distributes over union.
Proposition 37. For each -object and -morphism , there is an involution on defined as follows:(i), where for all ;(ii) is the converse M-soft set relation , where .
Proof. Let , and . Since is the converse -soft set relation of , we have . In addition, is a SRel-morphism, so for every and , which means that . It is immediate that is a SRel-morphism from to . Furthermore, assume that is a -soft set relation from to ; then by Definition 29, we can easily obtain that , whence is compatible with composition. At last, obviously, takes the identity map on to the identity map on . Hence, is a contravariant functor that is obviously period two. By Definition 7, is an involution on SRel.
Remark 38. It should be noted that the notion of involution gives a bijective mapping from homset to .
Proposition 39. Let be a -morphism from to . Then the following statements are equivalent:(i) is monic;(ii)if , then the map , defined by , is one-one;(iii)for every , there exists such that is the only element related to .
Proof. (i)(ii) Suppose that and . Take a singleton and assume that the map sends to . Define two relations from to by setting and . As is the subset of any sets, we have being SRel-morphisms and for each . Since , by the definition, for each , there exist and such that if and only if . It is immediate that there exist such that and if and only if and . By Definition 29, . Since is monic, we have . Consequently, , which means that is one-one.
(ii)(iii) Assume that is one-one; then . It follows from the definition that for each there exists such that is the only element related to .
(iii)(i) Let and ; then we claim that there exist and such that , but . By (iii), take with , and then it follows from Definition 29 that , but does not hold, which means that . Hence is monic.
Proposition 40. Let be a -morphism from to . Then the following statements are equivalent:(i) is epic;(ii)the mapping , defined by , is one-one;(iii)for every , there exists with being the only element related to .
Proof. The proof is similar to that of Proposition 39.
Lemma 41. Let be a map defined by for each . Then is injective in .
Proof. For each SRel-objects , , let be monic and . Assume that . Since is monic, it follows from Proposition 39 that for every there exists such that is the only element related to . Define by . Apparently, is a SRel-morphism. According to the definition of , we can infer that . That is, the following diagram
is commutative. By Definition 8, is injective.
Theorem 42. Let be a -object. Then the identity embedding is an injective hull.
Proof. By Proposition 39, is monic. In addition, according to Lemma 41, is injective. Assume that such that is monic. Because of a relation rather than a morphism, , whence Proposition 39 gives that is monic. It follows from Definition 8 that is an injective hull.
Remark 43. It is well known that projective objects of a category are dual to that of injective objects and projective covers are dual to that of injective hulls (see ). So we can easily obtain the following proposition.
Theorem 44. Let be a mapping defined by for every . Then is the projective object of . Further, for each -object , the mapping is a projective cover of .
5. Adjoint Situations
We in this section mostly consider the relations among the categories SFun, Set, SRel, and Rel. In particular, we investigate the essential connections of SFun and SRel by means of adjoint situations.
Definition 45. Let be a forgetful functor which sends an object to and sends a morphism to .
Analogously, we can define another forgetful functor .
Definition 46. Define and for an object and morphism by setting(i) and to be the object , where defined by for all ;(ii) and to be the object , where defined by for all ;(iii), and to be , where is considered with the appropriate domain and codomain.
Theorem 47. (i) The pair is an adjoint situation.
(ii) The pair is an adjoint situation.
(iii) The pair is an adjoint situation.
(iv) The pair is an adjoint situation.
Proof. We just prove (i) and (ii) because the proofs of (iii) and (iv) are similar.
(i) Let be Rel-objects and SRel-object. By Definition 46(i), for all and ; we can infer that a morphism in Rel will lift to a morphism in SRel, whence . According to Definitions 45 and 46(i), . It follows from Definition 9 that is an adjoint situation.
(ii) Assume that are Rel-objects and is a SRel-object. By Definition 46 (ii), for every and , which means that a Rel-morphism can lift to a SRel-morphism from to . Thus . By Definitions 45 and 46 (ii), . Therefore, is an adjoint situation according to Definition 9.
Theorem 48. Assume that is the inclusion functor and define a functor where . Then is an adjoint situation.
Proof. Firstly, we show that is well defined. Let be a SRel-morphism. For , where is defined like Proposition 39, there exists such that . Since for every , we have , which implies that is a SFun-morphism. That is, the definition of is reasonable. Secondly, we claim that is an adjoint situation. It remains to show that . In fact, if is a SRel-morphism, then for all and , which implies that . That is, is a SFun-morphism. Consequently, assume that is a SFun-morphism. Define a relation by if . Then implies , whence and is a SRel-morphism.
Definition 49. Assume that are defined by and . Then and are called the projective cover and injective hull functor, respectively.
Proposition 50. Consider
Proof. It is straightforward from Theorems 42 and 44.
Soft set theory, a new powerful mathematical tool for dealing with uncertain problems, has recently received wide attention in both the real-life applications and the theory studies. In recent years, the combination of soft set theory and category theory has resulted in many interesting research topics. In this paper, we mostly focus on offering theoretical results by combining soft set theory and category theory. In other words, we have provided a categorical viewpoint for soft set theory and the results given in this paper can further enrich soft set theories. Particularly, we have proved that the category SFun is Cartesian closed, which will provide an important theoretical background for theoretical computer sciences. Naturally, applying our results to other fields such as information sciences and logic is also a valuable work and we will present it in the future work.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (Grant no. 11071151) and the Special Fund of Shaanxi Provincial Education Department (Grant no. 2013JK0568).
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