A Novel Concept of Fuzzy Subsemigroup (Ideal)

Che, Yu-Hong; Liu, Qi

doi:https://doi.org/10.1155/2022/5830590

Journal of Mathematics

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 5830590 | https://doi.org/10.1155/2022/5830590

A Novel Concept of Fuzzy Subsemigroup (Ideal)

Yu-Hong Che¹and Qi Liu²

Academic Editor: Feng Feng

Received28 May 2022

Revised16 Aug 2022

Accepted08 Sept 2022

Published30 Sept 2022

Abstract

This paper focuses on a generalized definition of fuzzy subsemigroup (ideal) on semigroup. Let be a completely distributive lattice; we introduce the definition of -fuzzy ideal and also the novel concept of subsemigroup (ideal) on semigroup. Then, we discuss the necessary and sufficient conditions of -fuzzy subsemigroup (ideal) measure using the four level cuts of an -fuzzy set. Moreover, we study the properties of -fuzzy subsemigroup (ideal) measure. As an application of -fuzzy subsemigroup (ideal) measure, we obtain the -fuzzy convexities on a semigroup and bijective semigroup homomorphic mapping is an -fuzzy isomorphism.

1. Introduction

Zadeh introduced fuzzy set theory [1]. Then, Van de Vel [2] gave a systematic study of the theory of convexity in 1993. Rosa [3] extended the concept of convexity to the fuzzy situation in 1994, which is said to be an -convex structure later, and the relevant results are shown in [4–8]. In addition, Maruyama [9] studied the concept of convexity on a completely distributive lattice in 2009. As a matter of fact, there is a convex space in many other mathematical structures [10–13].

Many academics studied on generalized semigroups (ideal) [14–19]. Zhao [19] introduced the concept of -fuzzy subsemigroup. To be worth mentioning, Shi and Xin opened up a prospect of inducing the -fuzzy convexity [20]. In recent years, mathematicians have constructed the -fuzzy convexities on a series of algebraic structures. Li and Shi [21] constructed an -fuzzy convexity according to -convex fuzzy sublattice measure. Furtherly, they obtained -fuzzy convexity spaces. In particular, an -fuzzy convexity became known simply as -fuzzy convexity if , where is a completely distributive lattice [22]. In 2017, Wen et al. [23] defined an -fuzzy convexity by the -convex measure on vector spaces. Two years later, Han and Shi [24] introduced an -fuzzy convexity according to -convex ideal measure on a lattice. At the same year, Zhong et al. [25] constructed an -fuzzy convexity according to -fuzzy convex subgroup measure on an ordered group. The study by Mehmood et al. [26] constructed an -fuzzy convexity on a ring in 2020. In 2021, Zeng et al. (preprint [27]) constructed an -fuzzy convexity according to an -fuzzy subfield degree on a field.

As a continuation of [19], we will extend fuzzy subsemigroup (ideal) on semigroup to general cases on the basis of the above researches. It recalled some necessary concepts and theorems of general semigroup, fuzzy semigroup, and -fuzzy convexity. Following that, the -fuzzy ideal is studied and its characterizations are given. Then, an -fuzzy subsemigroup (ideal) measure is presented, and at the same time, the characterizations and properties are studied. Then, an -fuzzy convexity is generated by an -fuzzy subsemigroup (ideal) measure on a semigroup. What is more, the related -fuzzy convex preserving mappings, -fuzzy convex to convex mappings, and -fuzzy isomorphism mappings are discussed. The findings show that a semigroup homomorphic mapping can be considered as -fuzzy convexity preserving (-fuzzy convex-to-convex) mapping and bijective semigroup homomorphic mapping is an -fuzzy isomorphism.

2. Preliminaries

To begin with, we give some useful materials for necessary. From [28], we know the operation of is defined as , where is a semigroup and .

Definition 1 (see [28]). Let be a semigroup. A nonemptysubset of is said to be a subsemigroup (ideal) of if

Moreover, the intersection of any collection of subsemigroup of is a subsemigroup of [28]. Beside, a semigroup is said to be a commutative semigroup if the operation of satisfies the commutative law.

Mordeson et al. extended Definition 1 to the fuzzy situation.

Definition 2 (see [28]). Let be a semigroup. An -fuzzy subset is called to be (1)a fuzzy subsemigroup of if such that (2)a fuzzy left (right) ideal of if such that (3)a fuzzy ideal of if it is a fuzzy left ideal and it is a fuzzy right ideal of

Throughout this paper, and denote the minimum element and the maximum element in , respectively. We denote as a subset of by , the set of all -fuzzy subsets on by , and an -fuzzy subset of by . is a completely distributive lattice if it maintains the lattice via defining point wisely, naturally. In addition, symbols and denote the minimum element and the maximum element in , respectively. Moreover, and . Let , and be three elements in . Then, is said to be a coprime if is less than or equal to leading to less than or equal to or less than or equal to . Moreover is said to be prime if is less than or equal to leading to either less than or equal to or less than or equal to . In addition, and denote the collection of all nonunit prime and nonzero coprime elements, respectively. and , less than or equal to , means that is less than or equal to for some . Then, is equal to and is equal to mainly based on [29]. Moreover, , , in another way, there exists in , by , less than or equal to ; we can have which is less than or equal to ; then, we say is less than or equal to and is less than or equal to [29]. Moreover, and , it means [23].

The four cut sets play an important part in fuzzy theory. Supposing and , according to the definition of cut sets and the fact that an implication operator in correspondence to in [23], we infer is equal to and [23]

We denote -fuzzy convexity and -fuzzy convexity preserving mapping (-fuzzy convex-to-convex mapping) with and , respectively.

From the definition of -fuzzy convexity [22], we know the following.

Definition 3 (see ([23], [28]). A mapping is called to be an LFC on if the following conditions are true: (1)(2) if is nonempty(3) if is nonempty and totally ordered

From [23], we know the following.

Definition 4 (see ([23]). Let and be spaces. A mapping is called to be an () if Then, is called to be an -fuzzy isomorphism if is bijective, , and .

In this paper, we aim at a novel definition of fuzzy subsemigroup (ideal) on semigroup. According to it, any fuzzy set can be considered as a fuzzy subsemigroup (ideal) generalized. This paper is arranged as follows: first, we will treat the novel definition of fuzzy subsemigroup (ideal) on the basis of the notion of an -fuzzy subsemigroup (ideal) on semigroup, their properties, and their equivalent characterizations. Then, we will obtain -fuzzy convexities induced by -fuzzy subsemigroup (ideal) measure and a conclusion that bijective semigroup homomorphic mapping is an -fuzzy isomorphism and so on.

3. A Novel Definition of an -Fuzzy Subsemigroup (Ideal)

In this section, the novel definitions of -fuzzy subsemigroup are introduced and their characterizations are given. First, we will extend Definition 2 to the -fuzzy situation. (1)Let and . We define as follows:

Definition 5. Let be a semigroup, ; then, is called to be (1)an -fuzzy subsemigroup of , if [19](2)an -fuzzy left (right) ideal of , if (3)an -fuzzy ideal of , if it is an -fuzzy left ideal and it is an -fuzzy right ideal of

Example 1. Let be a semigroup with the following table.

From Definition 6 (1), we can infer that is an -fuzzy subsemigroup of , but is not. (2)Then, we define as follows:and as follows:

From Definition 5 (2), we get is an -fuzzy ideal of , but is not.

According to the definition of an -fuzzy subsemigroup in [19], an -subset is either an L-fuzzy subsemigroup or not. We discuss the measure to which an -subset is an -fuzzy subsemigroup using the implication operator of . Moreover, we define an -fuzzy ideal and study the measure to which an -subset is an -fuzzy ideal.

Definition 6. Let be a semigroup, ; then, (1) is called to be the subsemigroup measure of as(2) is called to be the ideal measure of as

Obviously, is an -fuzzy subsemigroup (ideal) of if and only if , and always is an -fuzzy subsemigroup (ideal) to the measure .

From Example 1, we know is an -fuzzy subsemigroup (ideal) of , but is not. We will explain -fuzzy subsemigroup (ideal) measure according to the following example, which indicates that any -subset can be regarded as an -fuzzy subsemigroup (ideal) to some measure.

Example 2. On the basis of Example 1, we define -fuzzy subsemigroup (ideal) measure of as

According to Definition 6 (1), this means that is an -fuzzy subsemigroup to the measure . In addition,

From Definition 6 (2), we can say that is an -fuzzy ideal to the measure .

By Definition 6, we can infer the following conclusion.

Theorem 7. Let be a semigroup and , then .

Inspired by the conclusion if and only if [23] , the following lemma is clear.

Lemma 8. Let be a semigroup and ; then, (1) if and only if (2) if and only if , and

By Lemma 8, we can easily obtain the other characterizations of the subsemigroup measure of an -fuzzy subset as follows.

Theorem 9. Let be a semigroup and ; then, (1)(2)

Next, we characterize the subgroup (ideal) measure of an -fuzzy set by means of its four levels in the following theorem.

Theorem 10. Let be a semigroup, ; then, (1)(2)(3)(4)(5)(6)(7)(8)

Proof. (1)First, we prove, let ; then, , ; we have This shows ; therefore, is a subsemigroup of . Thus, Next, we prove Let be a subsemigroup of , , . We want to show Suppose , we can infer and , then if and only if . Thus, Now, we get (2)First, we proveWe take any , if , then , if and only if ; we can obtain By , then we can infer if and only if . This means is a subsemigroup of , and This demonstrates Next, we prove Suppose is a subsemigroup of for any . We want to show Suppose , then and ; thus, if and only if . This shows . It is proven that Thus, (2) is true. (3)First, we proveSupposing , it satisfies . Let , , and ; now we prove Assume that According to the fact if and only if , we can infer . By and , now, we can get , which contradicts . Hence, . This shows that is a subsemigroup of . Therefore, Next, we prove Supposing , it satisfies . Now, we want to prove Suppose and , then , , and if and only if . For a subsemigroup , it means if and only if . This shows that . It is proved that , It is now obvious that (3) holds. (4)First, we proveWe assume , , then , ; it holds that Then, . This shows that is a subsemigroup of . This means that Next, we prove Supposing , it satisfies . Now, we prove that Let . By , we know that , and . Since is a subsemigroup of , it holds that if and only if . This shows that . Therefore, It is now obvious that (4) holds.
(5)-(7) By the fact that , , and are the ideals of for any and , then the statements are easy to be proved with Lemma 8 (2) and are omitted (8)Suppose and . It is a fact that if and only if and . In addition, this shows that is an ideal of . This means that for any , ,Conversely, suppose is an ideal of . Now, we prove that Let ; by , we know that and . For an ideal of , it holds that if and only if ; it shows that . We can obtain in a similar way. Therefore, It is now obvious that (8) holds.

Theorem 11. Let be a family of -fuzzy subsets in a semigroup . Then, (1)(2)

Proof. Suppose is a family of -fuzzy subsets in a semigroup . (1)Suppose , then . We can infer (2)Suppose , then ; it is easy to obtain , and by , we can obtainWe can infer .

The following result is obvious.

Let be a mapping and and be two -fuzzy subsets in and , respectively. Then, we have (1) if is surjective(2) if is injectivewhere and are defined by

Now, we investigate the subsemigroup (ideal) measures of homomorphic image and preimage of -fuzzy subset.

Theorem 12. Suppose is a semigroup homomorphism. Then, (1)if , then . And if is injective, then (2)if , then . And if is surjective, then (3)if , then . And if is injective, then (4)if , then . And if is surjective, then

Proof. (1)We complete the proof by combining the factIf is injective, the above can be replaced by , i.e., . Now, (1) is true. (2)We complete the proof by combining the factIf is surjective, the above can be replaced by , i.e., Now, (2) is true. (3)We complete the proof by combining the factFor any If is injective, the above can be replaced by , i.e., Now, (3) is true. (4)We complete the proof by combining the factIf is surjective, the above can be replaced by , i.e., It is now obvious that the theorem holds.

On the basis of the definition of the Cartesian product of the -fuzzy subset [20], we discuss the subsemigroup measure of the product of -fuzzy subsets of a semigroup. (2)The proof is simple, and we omit it

Theorem 13. Let be a family of semigroup and be the product of , where . Then, (1)(2)

Proof. (1)For , let be projections ; then, is semigroup homomorphic mappings; we can inferThen,

Definition 14. Let be a semigroup, , and , then defined by It is obvious that .

Example 3. Let be a semigroup with the same operation as Example 1. We define as follows: From Definition 14, we get

Theorem 15. Let be a semigroup, . If , then

Proof. . Let for any , then and . Then, by Theorem 10 (3), and are subsemigroups, respectively. By , then is a subsemigroup of . We can infer . It is now obvious that the theorem holds.

Theorem 16. Let be a commutative semigroup, , , then

Proof. Suppose for any ; we can infer and . Then, ; by Theorem 10 (7), and are an ideal, respectively. By , then is an ideal of . We can infer . It is now obvious that the theorem holds.

4. The -Fuzzy Convexities and Their Properties

In the following, we will use the -fuzzy subsemigroup (ideal) measures, we will construct an -fuzzy convexity on a semigroup. After that, we will study the relationship between and ().

, () can be reasonably considered as a mapping : : specified by . It will indicate that is still an on a semigroup .

The following convention is that the null set is a subsemigroup (ideal) of any semigroup. It is necessary to discuss the next theorem.

Theorem 17. Let be a semigroup, . Then, (1)the mapping : specified by is an induced via -fuzzy subsemigroup measure on (2)the mapping : specified by is an induced via -fuzzy ideal measure on

Proof. (1) We will prove (1) in accordance with the three axioms of Definition 3 as follows: (i)It indicates (ii)Theorem 11 has proven if , a family of -fuzzy subsets in a semigroup , is nonempty(iii)Suppose , is a chain. We want to proveIt needs to show that By Lemma 8( 2), , we know Assume ; it satisfies Then, we have So there exists ; it satisfies For , suppose ; it follows that . By , we obtain . Hence, . Then, we can infer Combining Lemma 8 (2), we have . Then, we obtain. Therefore, it is proved.
(2) There is something in common shown in Theorem 17 (1), and it is omitted here.

In the rest of this paper, we will study the properties of on semigroup. We denote -convexity with . From [22, 23], we get the equivalent characterization between an and . It is found that is an if and only if is an for any or is an for any . Then, we get the next theorem.

Theorem 18. Let be a semigroup; then, the following characterizations are equivalent: (1) is the induced by , where (2), is an on (3), is an on

Combining Definition 3 and Theorem 12, we can get the next theorem.

Theorem 19. Let be a semigroup homomorphic mapping and and be the -fuzzy convexities induced by -fuzzy subsemigroup (ideal) measures on and , respectively. Then, (1) is an (2) is an (3) is an (4) is an

Proof. To prove (1) and (2) hold, combining Theorem 12 (2) and (4) with the conditions is a semigroup homomorphic mapping and and are the -fuzzy convexities induced by -fuzzy subsemigroup (ideal) measures on and , we can get Then, we can infer (1) and (2) hold according to Definition 4.
Similarly, we can prove (3) and (4) hold.

Combining Theorem 13 and Theorem 19 with Definition 4, we can get the next theorem in a similar way of Theorem 19.

Theorem 20. Let be a family of semigroup and be a family of projection by . Then, (1) () is an (2) () is an

From Definition 4 and Theorem 19 again, we can infer the following theorem.

Theorem 21. Let be a semigroup homomorphic mapping and a bijective by . Then (1) is an -fuzzy isomorphism(2) is an -fuzzy isomorphism

5. Conclusions

It is presented in this paper that any -fuzzy subsets can be considered as an -fuzzy subsemigroup (ideal) on semigroup to some measure. By this, we obtained -fuzzy convexities and studied their properties. Next, we will research the related definitions and properties on order semigroup.

Data Availability

All data supporting the findings of this study are included in the submitted article.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

Y.-H.C. and Q.L. were responsible for conceptualization, software, and formal analysis; Q.L. was responsible for the methodology, supervision, project administration, and funding acquisition; Y.-H.C. was responsible for writing the original draft preparation, writing, review, and editing. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

The authors would like to thank Professor Fu-Gui Shi for his helps and suggestions. The project is funded by the National Natural Science Foundation of China (12101210) and the Scientific Research Project of Hubei Provincial Department of Education (Q20201903).

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Copyright © 2022 Yu-Hong Che and Qi Liu. 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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