Gradation of Continuity for Fuzzy Soft Mappings

Çetkin, Vildan

doi:https://doi.org/10.1155/2021/2691753

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Research Article | Open Access

Volume 2021 | Article ID 2691753 | https://doi.org/10.1155/2021/2691753

Gradation of Continuity for Fuzzy Soft Mappings

Vildan Çetkin¹

Academic Editor: Ljubisa Kocinac

Received05 Jul 2021

Accepted07 Aug 2021

Published19 Aug 2021

Abstract

This paper is devoted to describe the notion of a parameterized degree of continuity for mappings between -fuzzy soft topological spaces, where is a complete De Morgan algebra. The degrees of openness, closedness, and being a homeomorphism for the fuzzy soft mappings are also presented. The properties and characterizations of the proposed notions are pictured. Besides, the degree of continuity for a fuzzy soft mapping is unified with the degree of compactness and connectedness in a natural way.

1. Introduction

The theory of fuzzy sets which is a way of modeling real-life problems involves uncertainty, based on the degree of membership of an element to some sets. This idea has impressed many researchers working in diverse areas. Especially, the topology workers applied this idea to the gradation of openness and hence gave birth to the fuzzy topology [1]. The fuzzy-fuzzy case of the topology is the most compatible way of reflecting the gradation of belongingness [2]. So naturally, the notions of the degree of compactness, the degree of connectedness, the degree of separations, and so on have been considered. Later on, the similar argument has been considered for the mappings between fuzzy topological spaces and the degree of continuity, the degree of openness, and the degree of closedness for (fuzzy) mappings have been described [3–5]. The theory of soft sets is one of the other tools to model vague phenomena [6]. Also, the combination of fuzzy sets and soft sets gave birth to the fuzzy soft sets [7–9] and the basic idea of this new kind of sets depends on the parameterized degree of membership of an element to some sets. Nowadays, the studies depending on the soft sets and the fuzzy soft sets are increasing rapidly [10–13].

The idea of fuzzy softness (in fact, “parameterized gradation”) is one of the appropriate tools for modeling of environmental and mathematical problems. On the other hand, the mappings play the key role in transforming the characteristics between structured spaces. Especially, the continuous mappings are worth to investigate since they preserve the several properties of the spaces endowed with topology. Motivated from this thinking, we found it reasonable to present a new theory which gives a more accurate and efficient way of transforming the characteristics between the fuzzy soft topological spaces depending on the parameters. Thus, as a continuation of the research studies [14–16], we describe the parameterized gradations of continuity, openness, and closedness for mappings between fuzzy soft topological spaces.

The content of this study is organized in the following manner: in Section 2, we present the notations and recall the elementary notions which are used throughout the study. In Section 3, we define the parameterized degree of the concepts of continuity, openness, closedness, and being homeomorphism for mappings transformed between fuzzy soft topological spaces. We investigate the parameterized graded extensions of the main properties and results known in general topology, for the proposed concepts. Additionally, we observe several characterizations of the described gradations with the help of the neighborhood systems and closure (interior) operators. At the end, we unify the parameterized graded continuity with the parameterized compactness (connectedness, respectively) degree.

2. Preliminaries

Throughout this paper, refers to a nonempty initial universe, denotes the arbitrary nonempty sets viewed on the sets of parameters, and denotes a complete De Morgan algebra with the smallest element and the largest element . With the underlying lattice , a mapping is said to be an -fuzzy set on and by , we denote the family of all -fuzzy sets on .

An element in is said to be coprime if implies that or . denotes the collection of all coprime elements of . We say is way below (wedge below) , in symbols, , if for every directed (arbitrary) subset , implies for some . Clearly, if is coprime, then if and only if . Details for lattices can be found in [17].

The binary operation in the complete De Morgan algebra is given by .

For all and , the following properties are satisfied:(1) iff (2) iff (3) and (4)(5) and (6)

Definition 1 (see [18]). A mapping is called an -fuzzy soft set on . This means that is an -fuzzy set on , for each parameter . Hence, an -fuzzy soft set can be considered as the parameterized extended version of an -fuzzy set. Intuitively, by a fuzzy soft set, one can describe the parameterized degree of belongingness.
From now on, we use the symbol to denote the collection of all -fuzzy soft sets on .

Definition 2. (see [18, 19]). Let be two -fuzzy soft sets on ; then, the set operations are defined as follows:(1) is an -fuzzy soft subset of and written by if , for each . and are called equal if and .(2)The union of and is an -fuzzy soft set , where , for each .(3)The intersection of and is an -fuzzy soft set , where , for each .(4)The complement of is denoted by , where is a mapping given by , for each . Clearly .

Definition 3. (see [18]) (1)An -fuzzy soft set on is called a null (or empty) -fuzzy soft set and denoted by , if , for each .(2)An -fuzzy soft set on is called an absolute (or universal) -fuzzy soft set and denoted by , if , for each . Clearly and .

Definition 4. (see [20]). Let and be a function. Then, the -fuzzy soft set defined as follows is called an -fuzzy soft point and denoted by .An -fuzzy soft point is said to belong to an -fuzzy soft set and denoted by if , for each .
The set of all nonzero coprime elements of is denoted by . It is noted that is exactly the set of all -fuzzy soft points.

Definition 5. (see [21]). Let and be two functions, where and are parameter sets for the crisp sets and , respectively. Then, the pair is called an -fuzzy soft mapping from to :(1)The image of under , denoted by , is an -fuzzy soft set on defined by(2)The inverse image of under , denoted by , is an -fuzzy soft set on defined by(3)If and are both injective (surjective or bijective, respectively), then is said to be injective (surjective or bijective, respectively).(4)If and are soft mappings, then their composition is defined by .

Definition 6. (see [14]). A mapping which satisfies the following certain axioms is called an -fuzzy -soft topology on . (O1) , for each . (O2) , for each and for all . (O3) , for all and for each .Then, the pair is called an -fuzzy -soft topological space. The value is interpreted as the degree of openness of an -fuzzy soft set with respect to the parameter . So, the fuzzy soft topology can be thought as the gradation of parameterized degree of openness. Hence, the parameterized degree of closedness of a given -fuzzy soft set is described by using the complement operator .
Let and be -fuzzy -soft topologies on . We say that is finer than ( is coarser than ), denoted by , if for each .

Definition 7. (see [14]). Let be an -fuzzy -soft topological space and be an -fuzzy -soft topological space. Let and be crisp functions. Then, the fuzzy soft mapping is called continuous if for all and for all .

Definition 8. (see [20]). For a fixed fuzzy soft point , let the mappings satisfy the following axioms for each and : (FSQ1) (FSQ2) If , then (FSQ3) (FSQ4) Then, the collection of maps presented above is called an -fuzzy -soft quasi-coincident neighborhood (shortly, -nhood) system on . The value represents the parameterized degree of being -nhood of to the fuzzy soft point .

Proposition 1 (see [20]). Let be an -fuzzy -soft topology on . Define the mapping as follows: for each and ,

Then, the set of is an -fuzzy -soft -nhood system on , called induced -fuzzy -soft -nhood system by .

Definition 9. For a fixed fuzzy soft point , let the mappings satisfy the following axioms for each and . (FSN1) (FSN2) If , then (FSN3) (FSN4) Then, the collection of maps is called an -fuzzy -soft neighborhood (shortly, nhood) system on .

Definition 10. (see [16]). A mapping is called an -fuzzy -soft closure operator on if it satisfies the following axioms for each : (C1) , for all . (C2) for any . (C3) for any . (C4) . (C5) , where .The value is interpreted as the degree to which belongs to the parameterized closure of the fuzzy soft set .

Example 1. (see [16]). Let be the closure operator given in a parameterized -soft topological space . In this case, the mapping is defined in such a way thatsatisfies the conditions of Definition 10.

Theorem 1 (see [16]). Let be an -fuzzy -soft topology on . Then, the mapping defined byis an -fuzzy -soft closure operator on , which is called the -fuzzy -soft closure operator induced by .

Definition 11. A mapping is called an -fuzzy -soft interior operator on if it satisfies the following axioms for each :(I1), for all (I2) for any (I3) for any (I4)(I5), where , for all The value is interpreted as the degree to which belongs to the parameterized interior of the fuzzy soft set .

Theorem 2. Let be an -fuzzy -soft topology on , and let be the nhood system induced by . Define a mapping by

Then, the mapping is an -fuzzy -soft interior operator on , which is called the -fuzzy -soft interior operator induced by .

Definition 12. (see [22]). Let be an -fuzzy -soft topological space. Then, identify a mapping in such a way that in order to picture the parameterized compactness degree,In this case, the value is interpreted as the compactness degree of with respect to the parameter . So, is said to be compact -fuzzy soft set with respect to if .

Definition 13. (see [16]). Let be an -fuzzy -soft topological space. Then, identify a mapping by the following manner in order to describe the connectedness degree in such spaces:In this case, the value is said to be the connectedness degree of an -fuzzy soft set with respect to .

Theorem 3 (see [16]). Let be an -fuzzy -soft topology on . Then, one can characterize the parameterized degree of connectedness of an -fuzzy soft set in the following way:

3. Degree of Continuity for Fuzzy Soft Mappings

In this section, we identify the degrees of continuity, openness, closedness, and being a homeomorphism for a fuzzy soft mapping. Then, we study some of their characterizations by means of the q-nhood, nhood, interior, and closure operators. We also observe the elementary features of the proposed notions.

Definition 15. Let be the -fuzzy -soft and -soft topological spaces, respectively, and be a soft mapping. Then, we define the following(1)The parameterized degree of continuity for is as follows: for all , The value represents to which is continuous with respect to some parameters. Hence, the degree of continuity for is computed by the formula .(2)The parameterized degree of openness for is as follows: for all , The value represents to which is open with respect to some parameters. Hence, the degree of openness for is computed by the formula .(3)The parameterized degree of closedness for is as follows: for all , The value represents to which is a closed map with respect to some parameters. Hence, the degree of being a closed map for is computed by the formula .

Definition 16. Let be the -fuzzy -soft and -soft topological spaces, respectively, and be a bijective soft mapping. Then, the parameterized degree of being a homeomorphism for the soft mapping is identified by . Hence, .

Remark 1. (1)If the value , then it is seen that for each and for each is satisfied. This gives us the definition of continuity for the mapping . Analogously, if or , then (2) and (3) of Definition 15 are just the definitions of openness and closedness for the mapping , respectively.(2)If , and are the corresponding identity mappings, then is the identity soft mapping and .

Theorem 4. Let be the -fuzzy -soft topological spaces (where i = 1, 2, 3) and and be the soft mappings. Then, the following is satisfied:(1)(2)(3)(4)

Proof. (2) If we consider Definition 15, then we obtain the following inequality:for any .
The above fact witnesses the proof.
The other conditions are similarly proved.
As it is well known in general topology, the composition of two homeomorphisms is again a homeomorphism. Then, by the above theorem, we get the following result for the gradation of homeomorphism.

Corollary 1. Let be the -fuzzy -soft topological spaces (where i = 1, 2, 3) and and be two bijective soft mappings. Then, is satisfied.

Theorem 5. Let be the -fuzzy -soft topological spaces (where i = 1, 2, 3) and be a soft mapping. If the fuzzy soft mapping is surjective, then we get the following properties:(1)(2)

Proof. (1) , for each , is satisfied by the surjectivity of the soft mapping . Hence, we gainfor all .
Since is arbitrary, we obtain the claimed inequality .

Theorem 6. Let be the -fuzzy -soft topological spaces (where i = 1,2,3) and be a fuzzy soft mapping. If the fuzzy soft mapping is injective, then we get the following properties:(1)(2)

Proof. (1) By considering the definitions, we get the following:

Theorem 7. Let be the -fuzzy -soft and -fuzzy -soft topological spaces. If is a bijective soft mapping, then we have(1)(2)(3)

Proof. (1) Since the soft mapping is bijective, then for each and for each . By considering these facts, we gain the following:This witnesses the fact that(2) It is similarly proved to that of (1).
(3) In order to obtain the proof, we will consider the equalities and (by the injectivity property) for each . So,In addition, we have

Corollary 2. Let and be an -fuzzy -soft and -fuzzy -soft topological spaces, respectively. If is bijective, then the following characterizations are valid:(1)(2)(3)

Theorem 8. Let and be an -fuzzy -soft and -fuzzy -soft topological spaces, respectively. For any fuzzy soft mapping , the following characterizations are satisfied:(1)(2)(3)(4)

Proof. (1) By Proposition 1, implies . Hence, we may obtainIn order to prove the converse, consider the fact that , for each . Hence, implies . By these observations, we gain the following:This completes the proof.
(2) It is similar to that of (1).
(3) and (4) proofs are obtained by using Theorems 1 and 2.

Theorem 9. Let be the -fuzzy -soft and -fuzzy -soft topological spaces. For the fuzzy soft mapping , the following properties are satisfied:(1)(2)(3)(4)(5)(6)(7)(8)

Proof. (5) Let us first consider the following inequality:In order to obtain the converse inequality, we will use the following fact:We may also considerNow, let us choose an arbitrary which satisfiesTherefore, is satisfied for all and for all . By the implication operator properties, we get . For all with , we obtain . Then, there exist and such that and and also . This implies . Fromwe gain . So we get . Hence, we also haveSince the arbitrariness of , we gainSince , for all , we obtain the following inequality:Hence we obtain the desired result.
By using Theorems 1 and 2, and also by considering some similar discussion, one can prove the other claims of the theorem.

Theorem 10. Let and be the -fuzzy -soft and -fuzzy -soft topological spaces. For the fuzzy soft mapping , we have the following result:for all and .

Proof. Let us choose an arbitrary such that . By the below wedge operation property, we have thatHence for any and for any , we gainBy considering the implication properties, we haveIn order to complete the proof, it is necessary to show thatLet . Hence, we have the following facts:From the implication operator properties, we getTherefore, we obtain the following:This witnesses the desired fact .

Corollary 3. Let and be the -fuzzy -soft and -fuzzy -soft topological spaces. If the fuzzy soft mapping is surjective, then we have that .

Theorem 11. Let and be the -fuzzy -soft and -fuzzy -soft topological spaces. For the fuzzy soft mapping , we havefor all and .

Proof. Let us choose an arbitrary such that . By Theorems 3 and 4, the following implications are obtained:Hence, there exists which satisfy the conditions such that and also , for each . That is, there exists satisfying , and such that . And also, is satisfied for each . We conclude that from these observations,Since the coprime element is arbitrary, we gain the desired inequality for the parameter , .

4. Conclusion

As it is well known, in real life, nothing is described with the help of the 2-valued logic since there is no only black and white in nature. But the idea of fuzzy thinking, which gives some degrees to the phenomena reflect the facts more correctly. In the topological point of view, fuzzy logic, which is also related with quantum mechanics, was applied to the gradation of belongingness and it gave birth to the fuzzy topology. On the other side, the soft set theory which emphasizes the importance/necessity of the parametrization tool for adequate mathematical modeling of the natural facts is one of the preferred tools by the researchers [13, 23–26]. Besides, the combination of these types of sets, named as fuzzy soft set, is one of the adequate tools for modeling. In this respect, we deal with the gradation of the continuity of mappings between fuzzy soft topological spaces (here both of the sets and the axioms of the structure are all fuzzy soft). Since the mappings play the main role to establish the relations between structured sets, the idea proposed here helps us to obtain more appropriate and compatible results in such spaces. Despite the theoretical benefits of this method, it is not easy to find numerical examples in application. But this could be overcome by taking a unit interval instead of a lattice.

In relation with the research in this study, notice that soft continuity seems to be the natural tool to prove results more similar to Weierstrass’s celebrated theorem. For further research, we hope to investigate this idea and try to find reasonable results. Furthermore, we hope to extend the proposed methods to Pythagorean fuzzy uncertain environments [27] as an additional research.

Data Availability

The data used to support the findings of this study are cited at relevant places within the text as references and are also available from the corresponding author upon request.

Conflicts of Interest

The author declares no conflicts of interest.

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Copyright © 2021 Vildan Çetkin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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