A New Approach to Concavity Fuzzification

Alshammari, Ibtesam; Alghamdi, Azza M.; Ghareeb, A.

doi:https://doi.org/10.1155/2021/6699295

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Fuzzy Approximation and its Applications in Stabilization of Dynamic Systems

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Volume 2021 | Article ID 6699295 | https://doi.org/10.1155/2021/6699295

A New Approach to Concavity Fuzzification

Ibtesam Alshammari,¹Azza M. Alghamdi,²and A. Ghareeb^2,3

Academic Editor: Ahmed Mostafa Khalil

Received28 Nov 2020

Revised21 Dec 2020

Accepted24 Dec 2020

Published18 Jan 2021

Abstract

In this paper, we introduce a more general approach to the fuzzification of fuzzy concavity. More specifically, the degree of -fuzzy concavity is introduced and characterized as a generalization of -concave structure and -fuzzy concave structure. Based on that, the degree of -fuzzy concavity preserving and -fuzzy concave-to-concave of a function are defined. Some properties and relationships between the degree of -fuzzy concavity preserving and -fuzzy concave-to-concave functions are discussed.

1. Introduction

The concept of convexity is a fundamentally important geometrical property that plays a significant role in pure and applied mathematics. It can be sorted into concrete and abstract convexity. In this paper, we are mainly focusing on the abstract convexity. Fuzzification of the abstract convexity was initiated by Rosa [1–3] in 1994. He introduced fuzzy convex to convex and fuzzy convexity preserving functions and discussed some of their properties. To be worth mentioning, he developed the theory of fuzzy convexity by introducing its subspace, product, and quotient structures.

Many researchers have been involved in extending the notion of fuzzy convexity to the broader frame work of lattice-valued setting. In [4], Maruyama extended fuzzy convexity to -fuzzy setting framework, where is a completely distributive lattice. Jin and Li [5] proposed two functors between the categories of convex spaces and stratified -convex spaces, where is a continuous lattice. Both of the functors have been used to prove the embedding of convex spaces in stratified -convex spaces as a reflective and coreflective subcategory, where satisfies the multiplicative condition. Also, stratified -convex spaces, convex-generated -convex spaces, weakly induced -convex spaces, and induced -convex spaces are introduced and their relationships are discussed category-theoretically by Pang and Shi [6]. In 2016, Pang and Zhao [7] introduced the concept of -concave spaces, concave -neighborhood systems, and concave -interior operators which are the dual concepts of -convex spaces, convex -neighborhood systems, and convex -interior operators. The isomorphism between these categories and the category of -convex spaces are discussed and studied when is a completely distributive lattice with an order-reversing involution.

Shi and Xiu [8] initiated the concept of -fuzzifying convex structure as a new approach to convexity fuzzification. Recently, Shi and Li [9] generalized the classical restricted hull operators to -fuzzifying restricted hull operators and used it to characterize -fuzzifying convex structures. In [10], Wu and Bai discussed some properties of hull operator and introduced -fuzzifying JHC property and -fuzzifying Peano property. Further, Xiu and his colleagues [11–14] verified the categorical relationship between -fuzzifying convexity and other related spatial structures. Later, Shi and Xiu [15] introduced and characterized the notion of -fuzzy convexity as an extension to -convexity and -fuzzifying convexity. In the new structure, each -fuzzy subset can be regarded as an -convex set to some degree. -fuzzifying convex spaces and -fuzzy convex spaces have also been investigated in many studies [16–21].

In 1992, Šostak [22] introduced the concept of fuzzy category. In a fuzzy category, the potential objects and morphisms could be such to some degrees. Later, Kubiak and Šostak [23] fuzzified the category of -valued -fuzzy topological spaces. In [24], Zhong and Shi characterized the degree to which a function is an -fuzzy topology. Moreover, the degree to which an -subset is an -open set with respect to is studied. Further, the degrees of continuity, openness, closeness, and quotient of a function with respect to the -fuzzy topologies and , are given and their properties are characterized. Further, several kinds of continuity, compactness, and connectedness are generalized with their elementary properties to -fuzzy topological spaces setting based on graded concepts (see [25–30]). In [31], Ghareeb et al. studied the -fuzzy measurability in view of degree. Firstly, they generalized -fuzzy -algebra by presenting the degree of an -fuzzy -algebra with respect to a mapping . Moreover, they defined and discussed some special degrees such as the degree of -fuzzy measurable mapping, -fuzzy measurable-to-measurable mapping, isomorphic mapping, and quotient mapping with respect to mappings between two -fuzzy measurable spaces in detail.

The aim of this paper is to present the degree of -fuzzy concavity as an extension of -fuzzy concave structure. Moreover, we present the degree of -fuzzy concavity preserving functions and -fuzzy concave-to-concave functions. Some properties and relationship between the degree of -fuzzy concavity preserving and -fuzzy concave-to-concave functions are characterized.

2. Preliminaries

This section begins with some introductory material on -fuzzy convexity. In the sequel, refers to a finite set, both and denote completely distributive lattices. The zero and the unit elements in and are symbolized by , and , , respectively. By (resp. ), we refer to the directed (resp. codirected) subfamily of . For a completely distributive lattice , there exists a residual implication which is right adjoint for meet operation and given by

Moreover, the operation is defined by

The following lemma lists some properties of implication operation.

Lemma 1. (see [32]). For any , , , , and , we have the following statements:(1)(2)(3)(4), hence whenever (5), hence whenever (6)An element is said to be a prime element if leads to or . Also, is said to be coprime if leads to or . The collection of all nonunit prime and nonzero coprime elements in are symbolized by and , respectively.

The binary relation on is defined as follows: for , , if and only if for every subset , the relation always leads to the existence of with . The family is called the greatest minimal family of , symbolized by , and . Moreover, for , we define and . In a completely distributive lattice , there exist and for each , , and (see [33–35] for more details).

In [7], Pang and Zhao introduced the concept of -concave spaces, which is a dual concept of -convex spaces as follows.

Definition 1. (see [7]). An -concave structure on a nonempty set is a subset of such that(L1), (L2) for each (L3) for each The pair is called an -concave space. A function is called -concavity preserving if implies that , and is called concave-to-concave function if for each .
The following definition extends -concavity to -fuzzy setting.

Definition 2. A function is called an -fuzzy concavity on if satisfies the following statements:(C1)(C2), for every (C3), for every The pair is called an -fuzzy concave structure. For all , the value represents the degree to which is concave -subset. For any two -fuzzy concavities and on , we say is coarser than (i.e., is finer than ) if and only if , for every . For any two -fuzzy concave structures and , the function is said to be(1)An -fuzzy concavity preserving function if for any (2)An -fuzzy concave-to-concave function if for any Obviously, a -fuzzy concave structure can be viewed as an -fuzzifying concave structure, where . Moreover, an -fuzzy concave structure is called an -concave structure [7]. Further, when , the -fuzzy concave structure is called a -fuzzy concave structure. A crisp convex structure can be regarded as a -fuzzy convex structure.

Example 1. If an -fuzzy cotopology (see [36, 37]) satisfiesfor every , then is called saturated -fuzzy cotopology. The pair is called an Alexandroff -fuzzy cotopological space. It can be easily verified that Alexandroff -fuzzy cotopological space is an -fuzzy concave structure.
For each and , we have the following two cut sets:

Theorem 1. Let be a function. Then, the following statements are equivalent:(1) is an -fuzzy concave structure(2)For any , is an -concavity on (3)For any , is an -concavity on In the following theorem, we assume the existence of an order-reversing involution “′” with the completely distributive lattice , i.e., is a completely distributive DeMorgan algebra.

Theorem 2. (see [20, 38, 39]) .The closure (resp. hull) operator of an -fuzzy concave structure is defined byThen, for every and , achieves the following statements:(H1)(H2)If , then (H3)If , then (H4)Conversely, let be an operator achieving (H1)–(H4) and the function is given byThen, is an -fuzzy concave structure.

3. The Degree of -Fuzzy Concavity

In this section, we present and characterize the degree of -fuzzy concavity. Moreover, -fuzzifying concavity degree is presented and its characterizations are introduced.

Definition 3. Let be a function. Then, defined byis called an -fizzy concavity degree (i.e., the degree to which is an -fuzzy concavity on ).

Remark 1. (1)If , then , for each and for each . It is exactly the definition of -fuzzy concave structure on (see Definition 2). Moreover, is an -fuzzy concavity if and only if .(2)If in Definition 3, then is called an -fuzzifying concavity degree of .

Example 2. If is a function such that for any . By Definition 3, we get .

Lemma 2. Let be a function. For any , if and only if , , for each and for each .
The following theorem can be proved using the previous lemma.

Theorem 3. For the function , we have

Theorem 4. Let be a function. Then,

Proof. Let , , and for every and for every . For , , and , we have , , , and . Therefore, , , , and and hence , where denotes the right hand side of the equality.
Conversely, suppose that is an -concavity on for any . Let , then , , which means and . For any , let , then and . Thus, , i.e., . For any , let , then and . Thus, , i.e., . Hence, .

Theorem 5. Let be a function. Then,

Proof. Let , , and for each and for each . For any , , and , we have . Then,Thus, and . This shows and . Since and , we know and , which proves , . Hence, , where denotes the right hand side of the equality.
Conversely, suppose that for any , is an -concavity and , , i.e., and . Then, and , i.e., and . For , take any . By , we have , , for all , which means . Since is an -concavity, we have , i.e., . Then, , i.e., . Similarly, we can show that for each . Hence, .
The following corollary gives similar characterization of -fuzzifying concavity.

Corollary 1. Let be a function. Then,(1)(2)(3)An -fuzzy concavity degree can also be treated as a function given by . This function has the following property.

Theorem 6. Let be a family of functions. Then,

Proof. From Definition 3, we have

4. The Degree of Concavity Preserving and Concave-to-Concave Functions

This section presents the degree of -fuzzy concavity preserving and -fuzzy concave-to-concave functions and discuss their properties.

Definition 4. Given a function , -fuzzy concavity degree of , for any , we define by is called the degree to which is an -concave set with respect to (or the -concave set degree of with respect to ).

Remark 2. If , which means that is an -fuzzy concavity, then , which can be regarded as a generalization of .

Proposition 1. Given a function and -fuzzy concavity degree of , for each , denotes the -concave set degree of with respect to . Then,(1), (2), That is, if is regarded as a function defined by , then is an -fuzzy concavity on .

Proof. (1)Based on Definition 4, it suffices to prove that and . By Definition 3, we have(2)It is similar to (3).The following theorem characterizes -concave set degree.

Theorem 7. Let be a function and be -fuzzy concavity degree of . For each , denotes the -concave set degree of with respect to . Then,(1)(2)(3)

Proof. The proofs are similar to those of Theorems 3–5.

Definition 5. Given two functions and , let be a function. Then,(1)The concavity preserving degree of with respect to and is defined by(2)The concave-to-concave degree of with respect to and is defined bywhere and .

Theorem 8. Let , be two functions and let , refer to the -fuzzy concavity degrees of and , respectively. Then,(1)(2)(3)

Proof. (1)Since for each and , we have The proof of (1) is clear.(2)Suppose that for each . For any and , i.e., , we have . Thus, and , i.e., . By (1), hence , where refers to the right hand side of equality. Conversely, let . Then, and , i.e., . Thus, and , i.e., . This implies . Since is arbitrary, we have . From this result together with (1), we have .(3)Suppose that for each . For any and , i.e., , we have . Since is a - map, it follows thatThis implies and , i.e., . By (1), we have , where refers to the right hand side of the equality.
Conversely, take any . Bywe have and , i.e., . Thus, and , i.e., . This impliesSince is arbitrary, we have . From this result together with (1), we get .

Theorem 9. Given two functions and , if and denote the -fuzzy concavity degrees of and , respectively, then(1)(2)(3)

Proof. Similar to the proof of Theorem 8.

Proposition 2. If is a complete distributive DeMorgan algebra and is a function between two -fuzzy concavity structures and , thenwhere and are the corresponding hull operators.

Proof. Straightforward.

Proposition 3. Given five functions , , , , and , then(1)(2)

Proof. (1)Since for each , by Definition 5, we have(2)similar to the proof of (1).

Proposition 4. Given three functions , , and , let and . Then,(1)If is surjective, then .(2)If is injective, then .

Proof. (1)Since is surjective, we know for all . Then, Hence,(2)Since is injective, we have for each . Then, for all . Hence,

Definition 6. For any two functions and , if the function is bijective, then the isomorphism degree of the function is given by .

Theorem 10. For any two functions and , if the function is bijective, then and .

Proof. From the bijectivity of the function , we have for any . Thus,Therefore, . This completes the proof.

Proposition 5. Given a function , if is the identity function, then

Proof. Straightforward.

Proposition 6. Given three functions , , and , let and be two bijective functions, then .

Proof. Straightforward.

Lemma 3. For any two functions and , if the function is bijective, then(1)(2)

Proof. (1)From the bijectivity of the function , we get for any and for any . Thus, Therefore,(2)similar to (1).

Theorem 11. For any two functions and , if the function is bijective, then(1)(2)

5. The Quotient Degree of Functions between -Fuzzy Concave Structures

In this section, we endow the quotient functions with some degree and discuss the relationship with the degree of -fuzzy concavity preserving functions and the degree of -fuzzy concave-to-concave functions.

Definition 7. For any two functions and , if the function is surjective, then the quotient degree of the function with respect to and , denoted by , is given byThe proof of the following theorem is similar to the proof of Theorem 8.

Theorem 12. For any two functions and , if and refer to the -fuzzy concavity degree of and , respectively, and is a surjective function, then(1)(2)(3)

Theorem 13. For any two functions and , if is a surjective function, then .

Proof. Straightforward

Theorem 14. For any two functions and , if the function is surjective, then .

Proof. From the surjectivity of , for each . Then,

Theorem 15. For any three functions , , and , if the functions and are surjective, then(1)(2)

Proof. (1)(2)SincethenTherefore,This completes the proof.

6. Conclusion

In this paper, we presented the degree to which a function is an -fuzzy concavity on a nonempty set . Moreover, the degree to which an -subset is an -concave set with respect to was considered. Also, we defined the concavity preserving, concave-to-concave degree, and quotient degree for functions between -fuzzy concave structures. Their characterizations were given and the relationships among them were discussed. We think our results will be useful to consider many properties of concave structures under degree of -fuzzy concavity. We think that studying the topological properties of such degree is the most abstract and generalization of these properties, which leads to the results of previous studies as soon as the degree equals to . Thus, the fuzzification theory would be applied in a better and more generalized way.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors extend appreciation to the Deanship of Scientific Research, University of Hafr Al Batin, for funding this work through the research group project no. G–104–2020.

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Copyright © 2021 Ibtesam Alshammari et al. 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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