[Retracted] (m, n)-Ideals in Semigroups Based on Int-Soft Sets

Muhiuddin, G.; Alanazi, Abdulaziz M.

doi:https://doi.org/10.1155/2021/5546596

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Volume 2021 | Article ID 5546596 | https://doi.org/10.1155/2021/5546596

[Retracted] (m, n)-Ideals in Semigroups Based on Int-Soft Sets

G. Muhiuddin¹and Abdulaziz M. Alanazi¹

Academic Editor: Sami Ullah Khan

Received08 Feb 2021

Accepted22 Jun 2021

Published07 Jul 2021

Abstract

Algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, and topological spaces. This provides sufficient motivation to researchers to review various concepts and results from the realm of abstract algebra in the broader framework of fuzzy setting. In this paper, we introduce the notions of int-soft -ideals, int-soft -ideals, and int-soft -ideals of semigroups by generalizing the concept of int-soft bi-ideals, int-soft right ideals, and int-soft left ideals in semigroups. In addition, some of the properties of int-soft -ideal, int-soft -ideal, and int-soft -ideal are studied. Also, characterizations of various types of semigroups such as -regular semigroups, -regular semigroups, and -regular semigroups in terms of their int-soft -ideals, int-soft -ideals, and int-soft -ideals are provided.

1. Introduction

Soft set theory of Molodtsov [1] is an important mathematical tool to dealing with uncertainties and fuzzy or vague objects and has huge applications in real-life situations. In soft sets, the problems of uncertainties deal with enough numbers of parameters which make it more accurate than other mathematical tools. Thus, the soft sets are better than the other mathematical tools to describe the uncertainties. Aktaş and Çaǧman [2] show that the soft sets are more accurate tools to deal the uncertainties by comparing the soft sets to rough and fuzzy sets. The decision-making problem in soft sets had been considered by Maji et al. [3]. In [4], Maji et al. investigated several operations on soft sets. The notions of soft sets introduced in different algebraic structures had been applied and studied by several authors, for example, Aktaş and Çaǧman [2] for soft groups, Feng et al. [5] for soft semirings, and Naz and Shabir [6, 7] for soft semi-hypergroups.

Song [8] introduced the notions of int-soft semigroups, int-soft left (resp. right) ideals, and int-soft quasi-ideals. Afterthat, Dudek and Jun [9] studied the properties of int-soft left (resp. right) ideals, and characterizations of these int-soft ideal are obtained. Moreover, they introduced the concept of int-soft (generalized) bi-ideals, and characterizations of (int-soft) generalized bi-ideals and int-soft bi-ideals are obtained. Dudek and Jun [9] introduced and characterized the notion of soft interior ideals of semigroups. The concept of union-soft semigroups, union-soft -ideals, union-soft -ideals, and union-soft semiprime soft sets have been considered by [10]. In addition, Muhiuddin et al. studied the soft set theory on various aspects (see, for example, [11–21]). For more related concepts, the readers are referred to [22–31].

The results of this paper are arranged as follows. Section 2 summarises some concepts and properties related to semigroups, soft sets, and int-soft ideals that are required to establish our key results, while Section 3 presents the principle of int-soft -ideals. We prove that the int-soft bi-ideals are int-soft -ideals for each positive integer , but the converse is not necessarily valid. Then, we prove that the subset of the semigroup is -ideal of if and only if over is an int-soft -ideal over . Also, we prove that a soft set over is an int-soft -ideal over if and only if . Moreover, we characterize regular semigroups in terms of int-soft -ideals over . In this respect, we prove that a semigroup is -regular if and only if for each int-soft -ideal over . In Section 4, first, we present the idea of int-soft -ideal and -ideal over . After that, we obtain some analogues’ results to the previous section. Furthermore, we prove that a semigroup is -regular if and only if for each int-soft -ideal and for each int-soft -ideal over . At the end of this section, we provide the existence theorem for int-soft -ideal over and for the minimality of int-soft -ideal over . We also provide a conclusion in Section 5 that contains the direction for certain potential work.

2. Preliminaries

Let be a semigroup. For , is defined as . A subset of is called a sub-semigroup of if . A subset of is called a left (resp. right) ideal of if and is called an ideal of if is both left and right ideal of . A sub-semigroup of is called a bi-ideal of if .

Let be a universal set and let be a set of parameters. Let denote the power set of and let . A pair is called a soft set (over ) [32] if is a mapping. We denote the set of all soft sets over with parameter set by .

Let and be soft sets over . Then, is called a soft subset of if and , .

Let and be two soft sets. Then, for each , the union and intersection are defined as

For any two soft sets and of , the int-soft product is defined as

A soft set over is called an int-soft right (resp. Left) ideal over if for all . It is called an int-soft ideal over if it is both int-soft left and int-soft right ideal over . An int-soft sub-semigroup over is called an int-soft bi-ideal over if for all . The set of all int-soft left (resp. Right) ideals and int-soft bi-ideals over will be denoted by (resp. ) and .

More concepts related to our study in different aspects have been studied in [33–39].

For , the characteristic soft set over is denoted by and defined as

Let . Then, we have (1) and (2) .

The concept of -ideals of semigroups was introduced by Lajos [40] as follows. Let be a semigroup and be nonnegative integers. Then, a sub-semigroup of is said to be an -ideal of if . After that, the concept of -ideals in various algebraic structures such as ordered semigroups, LA-semigroups, and fuzzy semigroups had been studied by, for instance, Akram et al. [41], Bussaban and Changphas [42], Changphas [43], Mahboob et al. [44], and many others.

We denote by the principal -ideal, the principal -ideal, and the principal -ideal generated by an element of , respectively. They were given by Krgovic [45] as follows:

In whatever follows, , , and denote the set of all -ideals, -ideals, and -ideals of .

3. Int-Soft -Ideals

Definition 1. An int-soft sub-semigroup over is called an int-soft -ideal over iffor all .
The set of all int-soft -ideals over will be denoted by .

Example 1. Let . Define the binary operation on as follows.

Then, is a semigroup. Define aswhere such that . It is straightforward to verify that .

Lemma 1. In , .

Proof (straightforward).

Remark 1. In general, in a semigroup , .

Example 2. Let . Define the binary operation on as follows.

Then, is a semigroup. Define asThen, , , but because .

Theorem 1. Let . Then, .

Proof. Let . We haveLet . Now, we haveTherefore, .

Theorem 2. Let . Then, .

Proof. Let . Below are the cases we have: Case 1. If for some , then Case 2. If for some , then When and for and are used in previous cases. Case 3. If , and , then . Therefore,Hence, .
Let and . Then, implies . Therefore, . Thus, , as required.

Theorem 3. Let . Then, .

Proof. Let . If , then . In the other case, when , then there exist elements such that , and . As , there exist such that , and . It is easy to show that there exist such that, for any , we have , and . As , there exist such that , and . Similarly, there exist , such that, for , we have and . Now, we have For any , let . Since , we haveHence, .

Definition 2. A semigroup is called the -regular if, such that .

Lemma 2. If is -regular, .

Proof. Suppose that and . Since is -regular, for some . Therefore,as required.

Lemma 3. Let . Then, , and .

Proof. Let . As , we have

Theorem 4. is -regular , .

Proof. Let . Then, for some . We haveTherefore, .
Let . Since , so by Theorem 2, . Therefore, . It follows that , and so, is -regular.

Theorem 5. is -regular .

Proof. Suppose that is -regular and . Then, by Theorems 3 and 4, and . Hence, .
Suppose that . As , by Theorem 2, . Thus, by hypothesis, we haveTherefore, . By Lemma 1 in [4], . Thus, , as required.

Lemma 4. If and is an int-soft sub-semigroup over , such thatthen .

Proof. As is an int-soft sub-semigroup over , by Theorem 3, it is sufficient to show that . Now,Hence, .

Lemma 5. Let and . If or , then(1)(2)

Proof. When , then we haveIt follows that is an int-soft sub-semigroup over . Also, we haveThus, . Similarly, when , then . Similar to (1), it can be verified.

4. Int-Soft -Ideals and Int-Soft -Ideals

Definition 3. An int-soft sub-semigroup over is called an int-soft -ideal over iffor all .
An int-soft -ideal can be described dually.
Whatever follows, we denote the set of all int-soft -ideals and -ideals over by and .

Example 3. Let . Define the binary operation on as follows.

Then, is a semigroup. Define asIt is straightforward to verify that and .

Lemma 6. In , .

Proof (straightforward).

Remark 2. In general, .

Example 4. In Example 3, , but .

Definition 4. A semigroup is called the -regular (resp. -regular) if such that .

Lemma 7. The following assertions hold:(1)In -regular , (2)In -regular ,

Proof. Let . Since is -regular, so such that . Therefore, we haveHence, . (2). Similarly, this can be proved.

Lemma 8. Let . Then, (resp. ) the (resp. ).

Proof. Let . If , for some , then . If for each , then . Therefore,Hence, .
Let and . Then, implies . Therefore, . Thus, , as required.

Theorem 6. Let be any int-soft sub-semigroup over . Then, (resp. ) (resp. ).

Proof. It is similar to the proof of Theorem 3.

Lemma 9. Let be -regular, , and . Then, and .

Proof. Let . Then, . We haveso we obtain . Hence, .

Theorem 7. In , the following assertions are true:(1) is -regular , (2) is -regular ,

Proof. Let . Then, such that . Now, we haveTherefore, .
Take any . Since , . Therefore, . It follows that and so, is -regular. Similar to (1), (2) can be verified.

Theorem 8. The following assertions are true in :(1) is -regular , (2) is -regular ,

Proof. (1) Suppose that is -regular and . Then, by Theorems 7 and 6, we have and . Hence, .
Take . By Lemma 8, . By hypothesis . So, , and it follows that . Therefore, by Theorem 1 in [45], is -regular. Similar to (1), (2) can be verified.

Theorem 9. is -regular , , and .

Proof. Suppose that and . As is -regular, we haveBy Theorem 8 and Lemma 9, we have and . Therefore, . Also, . Therefore, .
Take and . By Lemma 2, and . By hypothesis, we haveit follows that . Thus, by Theorem 12 in [44], is -regular.

Corollary 1. If is -regular, then , and .

Theorem 10. is -regular and .

Proof. Suppose that and . As is -regular, we haveand so, . Similarly, . Thus, . Since and , the reverse inclusion holds. Hence, .
Take and . By Lemma 8, and . Observe that, by hypothesis, we haveand it follows that . Therefore, by Theorem 3 in [45], is -regular.

Lemma 10. For , (resp. ).

Proof (straightforward).

Lemma 11. In -regular semigroup , for each , there exist and such that .

Proof. Suppose that . Then, . As is -regular, . Therefore, . Let and . By Lemma 9, and . Since is -regular, and , soas required.

Lemma 12. In -regular semigroup , and , .

Proof. Let and . Now,Therefore, .
By Lemmas 11 and 12, we have the following.

Theorem 11. Let be a -regular and . Then, there exist and such that .

Definition 5. An int-soft -ideal over is called minimal if, for all int-soft -ideal over , implies .
Dually, a minimum int-soft -ideal and minimal int-soft -ideal over can be described.

Theorem 12. In -regular semigroup , a soft set over is a minimal int-soft -ideal over there exist a minimal int-soft -ideal and a minimal int-soft -ideal over such that .

Proof. Let be minimal. By Lemma 11, . We show that is minimal. To show this, let such that . Since is -regular, so, by Corollary 1, . Again, by Corollary 1, . By Lemma 12, . Since , by minimality of the int-soft -ideal over , we have . Therefore, . As , we have . So, . Hence, . Thus, is minimal. Similarly, is minimal.
Assume that for some minimal int-soft -ideal and minimal int-soft -ideal over . By Lemma 11, . To show that is minimal, let such that . Then, .
As and is minimal, . Similarly, . Now, . Hence, is minimal.

Corollary 2. There is at least one minimal int-soft -ideal over in -regular semigroup has at least one minimal int-soft -ideal and one minimal int-soft -ideal over .

5. Conclusion

The main purpose of this article is to present in semigroups the ideas of int-soft -ideals, int-soft -ideals, and int-soft -ideals. If we take in the int-soft -ideals, int-soft -ideals, and int-soft -ideals in particular, then we get the int-soft bi-ideals, int-soft right ideals, and int-soft left ideals. The ideas proposed in this paper can also be seen to be more general than int-soft bi-ideals, int-soft right ideals, and int-soft left ideals. Also, if we place in the results of this paper, then the results of [8] are deduced as corollaries, which is the main application of the results of this paper.

In the future work, one can further study these concepts to various algebraic structures such as semi-hypergroups, semi-hyperrings, rings, -semigroups, BL-algebras, MTL-algebras, R0-algebras, MV-algebras, EQ-algebras, and lattice implication algebras.

Data Availability

No data were used to support the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2021 G. Muhiuddin and Abdulaziz M. Alanazi. 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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