An Approach to the Concept of Soft Fuzzy Proximity
The purpose of this paper is to introduce the concept of soft fuzzy proximity. Firstly, we give the definitions of soft fuzzy proximity and Katsaras soft fuzzy proximity, and also we investigate the relations between the soft fuzzy proximity and slightly modified version of Katsaras soft fuzzy proximity. Secondly, we induce a soft fuzzy topology from a given soft fuzzy proximity by using soft fuzzy closure operator. Then, we obtain the initial soft fuzzy proximity from a given family of soft fuzzy proximities. So, we describe products in the category of soft fuzzy proximities. Finally, we show that a family of all soft fuzzy proximities on a given set constitutes a complete lattice.
In 1999, Molodtsov  proposed a completely new concept called soft set theory to model uncertainty, which associates a set with a set of parameters. Later, Maji et al.  introduced the concept of fuzzy soft set which combines fuzzy sets and soft sets. Soft set and fuzzy soft set theories have a rich potential for applications in several directions. The algebraic structure of soft set and fuzzy soft set theories dealing with uncertainties has been studied by some authors [3–10]. The topological structure of fuzzy soft sets based on the sense of Šostak  has also been studied by Aygünoğlu et al. . They proved that the category of fuzzy soft topological spaces is a topological category over SET3.
There are known some different approaches to the concept of a fuzzy proximity in the literature. The first and the most advanced one was developed on the whole by Katsaras [13, 14], the second due to Artico and Moresco [15, 16]. Although these approaches proceed from different starting points, both of them are consistent with Chang (or Lowen) fuzzy topologies. Markin and Šostak  introduced the different concept of a fuzzy proximity which is consistent with the notion of a fuzzy topology as it is defined in . They considered basic properties of these fuzzy proximities, described how a fuzzy proximity generates a fuzzy topology (in the sense of ), and discussed the interrelations between their approaches and each of the two abovementioned approaches to the concept of a fuzzy proximity. The notion of an -fuzzy preproximity spaces where is a strictly two-sided, commutative quantale lattice having a strong negation was introduced by Kim and Min  from a somewhat different point from that in . Double fuzzy preproximity was introduced and studied by Zahran et al. . Çetkin and Aygün  gave the definition of a lattice valued double fuzzy preproximity spaces as an extension of  and studied some of its structural properties.
Despite the existence of all these approaches to the concept of fuzzy proximity, we decide to generalize more natural definition of fuzzy proximity to the soft theory. In this paper, we give an approach to the concept of soft fuzzy proximity which is the extension of the fuzzy proximities studied in  to the soft theory. In the case of the parameter sets and that are both one-pointed sets, we obtain the results given in . Firstly, we define soft fuzzy proximity space and Katsaras soft fuzzy proximity space and also investigate the relations between these spaces. Secondly, we generate a soft fuzzy topology from a given soft fuzzy proximity by using the closure operator. Thirdly, we prove the existence of the initial soft fuzzy proximity to describe products in the category SFP of soft fuzzy proximity spaces and soft proximally continuous maps. Finally, we show that a family of all soft fuzzy proximities on a given set is a complete lattice.
Throughout this paper, refers to an initial universe, is the set of all parameters for , and is the set of all fuzzy sets on (where ). For , for all . A fuzzy point in is a fuzzy set , where , such that when and otherwise. A family of fuzzy points is denoted by .
Definition 2 (see ). Let and be two fuzzy soft sets on ; then(1)we say that is a fuzzy soft subset of and write if , for each . and are called equal if and ;(2)the union of and is a fuzzy soft set , where , for each ;(3)the intersection of and on is a fuzzy soft set , where , for each ;(4)the complement of a fuzzy soft set is denoted by , where is a mapping given by , for each ;(5) is called a null fuzzy soft set and denoted by , if , for each , ;(6) is called an absolute fuzzy soft set and denoted by , if , for each , .
Definition 3. A fuzzy soft point is a fuzzy soft set, where ; for all . In other words, a fuzzy soft point can be considered as a collection of fuzzy points with respect to the parameters. A fuzzy soft point is said to belong to a fuzzy soft set , denoted by if and only if or equivalently , for each .
Definition 4 (see ). Let and be two functions, where and are parameter sets for the crisp sets and , respectively. Then the pair is called a fuzzy soft mapping from to . Let and be two fuzzy soft sets on and , respectively.(1)The image of under the fuzzy soft mapping , denoted by , is the fuzzy soft set on defined by
(2)The preimage of under the fuzzy soft mapping , denoted by , is the fuzzy soft set on defined by
If and are injective (surjective), then is said to be injective (surjective).(3)Let be a fuzzy soft mapping from to and let be a fuzzy soft mapping from to . Then the composition of these mappings from to is defined as follows: , where and .(4)The image of the fuzzy soft point under the fuzzy soft mapping is defined as follows:
For more details about fuzzy soft sets and fuzzy soft mappings, please refer to [2, 4, 12, 21–23].
The second parameter set belonging to the context of our work is denoted by , which is the parameter set for the fuzzy soft topological structures.
So throughout this study, let and be arbitrary nonempty sets viewed on the sets of parameters.
Definition 5 (see ). A mapping is called an -soft fuzzy topology on if it satisfies the following conditions for each :(O1);
(O2), for all ;(O3), for all , .Then the pair is called an -soft fuzzy topological space. The value is interpreted as the degree of openness of a fuzzy soft set with respect to parameter .
Let and be -soft fuzzy topologies on . We say that is finer than ( is coarser than ), denoted by , if for each , .
Example 6 (see ). Let be a fuzzy topology on in Šostak’s sense; that is, is a mapping from to . Take and define as which is levelwise fuzzy topology of in Chang’s sense, for each . However, it is well known that each Chang’s fuzzy topology can be considered as Šostak fuzzy topology by using fuzzifying method. Hence, satisfies (O1), (O2), and (O3).
According to this definition and by using the decomposition theorem of fuzzy sets , if we know the resulting soft fuzzy topology, then we can find the first fuzzy topology. Therefore, we can say that a fuzzy topology can be uniquely represented as a soft fuzzy topology.
Example 7. Let be a parameter set, let be the set of natural numbers, and let be defined as follows: for each , It is easy to testify that is a soft fuzzy topology on .
Definition 8 (see ). Let be an -soft fuzzy topological space and let be an -soft fuzzy topological space. Let , and be functions. Then the mapping from into is called a soft fuzzy continuous map if for all , .
The category of soft fuzzy topological spaces and soft fuzzy continuous mappings is denoted by SFTOP.
Definition 9 (see ). A map is called an -soft fuzzy closure operator on if and only if satisfies the following conditions, for each , (where ) and :(C1);(C2);(C3)if , then ;(C4)if , then ;(C5).The pair is called an -soft fuzzy closure space. An -soft fuzzy closure space is called topological if it provides
Definition 10 (see ). Let be an -soft fuzzy closure space and, be an -soft fuzzy closure space. Let , and be functions. Then, a map from to is called a soft fuzzy -map if
3. Soft Fuzzy Proximity
In this section, we define soft fuzzy proximity structures in the sense of Šostak and Markin and Katsaras, respectively. Then we investigate their relations from the categorical point of view.
Definition 11. A mapping is called an -soft fuzzy proximity if it satisfies the following axioms for each (P1); (P2), for all ;(P3), for all ;(P4); (P5). A pair where is a set and is an -soft fuzzy proximity on it is called an -soft fuzzy proximity space. Also, we call a gradation of nearness between the fuzzy soft sets and according to the parameter . Here, for each , is a mapping from into .
Remark 12. It is easy to notice that if is an -soft fuzzy proximity on , then for each , . If and , then for all .
Definition 13. Let be an -soft fuzzy proximity space and let be an -soft fuzzy proximity space. Let , and be functions. Then the mapping from into is called a soft proximally continuous if
Lemma 14. Let and be an -soft fuzzy proximity space and an -soft fuzzy proximity space, respectively. Let be a mapping from into . Then, is soft proximally continuous if and only if The category of soft fuzzy proximity spaces and soft proximally continuous mappings will be denoted by SFP.
Definition 15. A mapping is called a Katsaras -soft fuzzy proximity on a set if for any and , the next conditions are satisfied:(K1) if and only if ;(K2) if and only if or ;(K3)if , then and ;(K4)if , then ;(K5)if , then there exists such that and .(Here, denotes the negation of ; we write as the synonym of . A pair is called a Katsaras -soft fuzzy proximity space, where for each , is a relation on .
A mapping where , are Katsaras -soft fuzzy proximity spaces (), respectively, is called soft proximally continuous if implies .
The category of Katsaras soft fuzzy proximity spaces and soft proximally continuous mappings will be denoted by KSFP.
It is obvious that KSFP can be identified as the full subcategory of SFP and additionally, the objects of this category satisfy the following axiom: Now, we give the generalization of pseudo-fuzzy proximity which is considered by Katsaras as a more restricted concept of a fuzzy proximity in his former papers [13, 14] to the soft case. (We use the expression “pseudo-fuzzy” because the “fuzziness” of this relation is rather a poor one: Katsaras pseudo-fuzzy proximities are in a canonical one-to-one correspondence with usual proximities ).
Definition 16. A mapping satisfying (K1), (K2), (K3), (K5), and the following strengthened version of axiom (K4) is called a Katsaras -soft pseudo-fuzzy proximity on : if for all ,
The corresponding full subcategory of KSFP will be denoted by KSFP. Obviously KSFP can be described as “intersection” of KSFP and SFP.
When studying fuzzy topologies (in the sense of ) it is often useful to apply the technique of representing a general fuzzy topology by means of a “continuous” decreasing system of its level Chang fuzzy topologies (see ). Our next aim is to obtain a similar result for soft fuzzy proximities. For this purpose we will use a slightly modified version of a Katsaras soft fuzzy proximity.
Definition 17. A mapping where is called an -Katsaras -soft fuzzy proximity on if it satisfies (K1), (K2), (K3), (K5), and the following axiom: for all , (where for each )
Proposition 18. Let be an -soft fuzzy proximity space and . For and let if and only if . Then is an -Katsaras -soft fuzzy proximity on .
Proof. The validity of axioms (K1), (K2), (K3), and (K5) for is obvious. If , then and hence holds for , too.
Let be an -soft fuzzy proximity space. For consider the system of -Katsaras soft fuzzy proximities on the set . It is easy to notice that this system is nonincreasing; that is, implies , and it is lower semicontinuous in the following sense: for each it holds .
Conversely, for each let a lower semicontinuous nonincreasing system of -Katsaras soft fuzzy proximities on a set be given. For each , define a mapping by the equality . Indeed, axioms (P1), (P2), and (P3) are obvious. To show axiom (P4) assume that ; then for each and hence by , for all . Since is arbitrary it follows that . To show axiom (P5) find such that and . Then and hence . Since is arbitrary, the validity of axiom (P5) follows from here.
Notice also that the -soft fuzzy proximity has exactly as its -Katsaras -soft fuzzy proximity: for each , . Indeed the inclusion is obvious. To verify the inverse inclusion assume that . Then for all , and, hence, by lower semicontinuity of , , too.
The obtained results can be gathered in the following statement.
Theorem 19. Let be an -soft fuzzy proximity space and for each , let . Then is a nonincreasing lower semicontinuous system of -Katsaras soft fuzzy proximities. Conversely, for each , given a family of nonincreasing systems of -Katsaras soft fuzzy proximities, by the formula , we can define an -soft fuzzy proximity on . Besides, if is lower semicontinuous, then .
Definition 20. Let , be -soft fuzzy proximities on a set . We say that is stronger (or smaller) than denoted by , if for each , . In this case we also say that is weaker (or larger) than . It is obvious that if and only if for each , , where and are the corresponding level -Katsaras soft fuzzy proximities.
Different from the situation in crisp mathematics where each category has a strictly determined class of morphisms, in fuzzy mathematics it is sometimes desirable to introduce a “measure of defectiveness” for “potential morphisms.” Below we define the defect of soft proximal continuity for a mapping of soft fuzzy proximity spaces.
Definition 21. By the defect of soft proximal continuity of a mapping from to , where is an -soft fuzzy proximity space and is an -soft fuzzy proximity space, we call the number Obviously, for each mapping . If is soft proximally continuous, then . If and are Katsaras soft fuzzy proximity spaces and is not proximally continuous, then . However, in case of general soft fuzzy proximity spaces may be any number in .
Proposition 22. If and are mappings of -soft fuzzy proximity spaces , , respectively, then .
4. Soft Fuzzy Topologies Generated by Soft Fuzzy Proximity
In this section, we generate a soft fuzzy topology from a given soft fuzzy proximity by using the closure operator.
Let be an -soft fuzzy proximity space, , and . The -closure of the fuzzy soft set (or the closure of at the level ) with respect to the parameter is defined by the equality .
Lemma 23. Consider (where and ).
Proof. To show , take and and choose such that . Then , and hence . Since obviously , we conclude that .
To show the inverse inequality take and . Besides, without loss of generality we may assume that , for all , (and not only , for all e , (and not only () and that . However this implies that . Hence, if for some , then . Thus, for each satisfying the inequality and hence .
Proposition 24. Let be an -soft fuzzy proximity space. Define the mapping by Then the mapping is a topological -soft fuzzy closure operator.
Proof. (C1) .
(C2) It is obvious that .
(C3) Let and . Since , and , then we have .
(C4) For , since , we have .
(C5) Applying axiom (P3), we get .
Hence, is an -soft fuzzy closure operator.
Let , and let be given. By (P5), . Then for each , there exists such that . Hence and . This implies . Hence . Here implies , for all . Then we have If we take limit for in the last step of the inequality, then we have . Hence the proof is complete.
Notation. Let and . It is easy to verify that for each , is a topology of fuzzy soft sets in the sense of Tanay and Kandemir .
Proposition 25. The -soft fuzzy closure operator is continuous along in the following sense:
Proof. Since for each , .
It follows that , for all . To prove the converse inequality it is sufficient to show that . Assume, contrary, that there exist , and such that . Then for each one can find such that and . Now, denoting we have and . The obtained contradiction completes the proof.
Proposition 26. For each and , and hence .
Proof. It is obvious that implies for each and hence . Take now and consider a decreasing sequence . Then for each . Applying Proposition 25, we get ; that is, .
From this proposition, similarly as in Theorem 2.6 , we have the following.
Theorem 27. The mapping defined by the equality, for each , is an -soft fuzzy topology on . Besides .
The -soft fuzzy topology constructed in Theorem 27 will be called generated by and, if necessary, will be denoted by .
Theorem 28. If a mapping is soft proximally continuous, then the mapping is soft fuzzy continuous.
Proof. Similarly to the proof of Theorem 2.7 in , it is sufficient to verify the inequality for each . From Lemma 23 it follows that . Since is soft proximally continuous it is and hence .
From Theorems 27 and 28 it follows that the procedure assigning to each soft fuzzy proximity generated soft fuzzy topology and leaving morphisms unchanged can be interpreted as a functor.
Corollary 29. By letting for every soft fuzzy proximity space and for every soft proximally continuous mappings , a functor from the category SFP into SFTOP of soft fuzzy topological spaces.
Proposition 30. If , are -soft fuzzy proximities on and , then .
Proof. It is easy and therefore omitted.
5. Initial Soft Fuzzy Proximity
The main aim of this section is to describe products in the category SFP. This description is based on the construction of initial soft fuzzy topologies given below, which has an interest of its own.
Definition 31. Let be a set, let be the parameter sets, let be a family of -soft fuzzy proximity spaces, and let , be a family of mappings. The weakest soft fuzzy proximity on for which all mappings , , are soft proximally continuous, is called the initial soft fuzzy proximity for this family of mappings.
The existence of the initial soft fuzzy proximity is provided by the next theorem, containing also its effective characterizations.
Theorem 32. Let be a set, let be the parameter sets, let be a family of -soft fuzzy proximity spaces, and let , be a family of mappings. Then the equality where and defines the initial for this family of mappings -soft fuzzy proximity .
Proof. Notice first that for each , (where ) if and only if for any finite covers , of and , respectively, there exist and such that for all . Besides, without loss of generality we can assume that .
Passing to the proof we shall first establish that thus defined is indeed an -soft fuzzy proximity. It is obvious that for each , and . To show that satisfies the third axiom notice first that and imply the inequality . Assume that and find covers , , , such that .
Besides, without loss of generality, we may assume that (otherwise take the cover , where ).
Over designating the fuzzy soft sets constituting the covers, for each let . Then and thus from the above inequality we get
From the right side of the inequality it follows that there exist and such that for all . Considering the two possible cases, and , we easily obtain contradiction with the left side of the inequality.
To show the fourth axiom assume that for some and find covers , and such that for all . It follows from here that . However, this is impossible, because is an -soft fuzzy proximity.
To establish the last axiom fix and consider the set of all pairs such that and . The validity of (P5) will follow from the fact that for each , is empty, in which we are going to establish.
Assume, contrary, that there exists and notice first that in this case for each . Indeed, take and let . If , then and hence . Similarly, if , then . Recalling the definition of and applying axiom (P5) to we conclude that .
For each there exist numbers and covers and such that for each pair one can find for which . Besides, we can assume the fuzzy soft sets , and their covers are chosen in such a way that the corresponding sum is the minimal one and that . Let . Then one of the following two possibilities should be true:(a)for every either or ;(b)for every either or .Indeed, assume that neither (a) nor (b) holds. Then there are such that , , , . Letting we get ; however, this contradicts the fact that .
Suppose that (a) holds. Then since and we conclude that and hence . However, this obviously contradicts the assumption of the minimality of the sum . In a similar way the case (b) can be excluded. Hence the set is empty.
Thus, is an -soft fuzzy proximity on . Besides, from its definition it follows that for all , and therefore all mappings are soft proximally continuous. Assume that there exists another -soft fuzzy proximity on such that and all are soft proximally continuous. then there exist , covers , , and such that for all . Since is an -soft fuzzy proximity, for some and hence . However, this is impossible because is soft proximally continuous.
Definition 33. By the product of -soft fuzzy proximity spaces , , we call the pair where , and are the product sets and is the initial -soft fuzzy proximity for the family of all projections , and , .
From Theorem 32 it follows that the product -soft fuzzy proximity can be defined by the formula It is easy to notice that the operation thus defined is indeed the product in the category SFP.
Theorem 34. Let , , be a family of mappings and let be the diagonal mappings. Then .
Proof. Consider the following: , and , .
Applying Definition 21 and noticing that we get and hence . To show the converse inequality assume that and find such that . Choose and realizing the right side of the inequality; then for any , and hence . Consider arbitrary covers , and take , satisfying . Then letting , in the above inequality we receive . Thus, for any covers , there exist , such that for any . Noticing that , and recalling the definition of the product soft fuzzy proximity we conclude that . The obtained contradiction completes the proof.
Corollary 35. The diagonal of a family of mappings is soft proximally continuous if and only if each is soft proximally continuous.
Another useful application of initial soft fuzzy proximities is given in the next theorem which describes lattice-theoretic properties of the family of soft fuzzy proximities.
Theorem 36. The set of all -soft fuzzy proximities on a given set is a complete lattice with respect to the order ≤.
Proof. Let be defined by for each , if and only if or and otherwise. Obviously is the largest (i.e., the weakest) -soft fuzzy proximity on ; it will be called antidiscrete. On the other hand let be defined by each ,