Abstract
A semigroup is called an -semigroup (-semigroup) if the set of all idempotents (the set of all regular elements) forms a subsemigroup. In this paper, we introduce the concept of -semigroups and establish the relationship between the three classes of semigroups.
1. Introduction and Preliminaries
Let be a semigroup. An element is called an idempotent if . An element is called regular if there exists such that . Denote by and the set of all idempotents and the set of all regular elements in , respectively. We call an -semigroup if forms a subsemigroup and a -semigroup if forms a subsemigroup. is called regular if and called orthodox if is a regular -semigroup. More details on regular semigroups and orthodox semigroups can be seen in [1]. For and , denote
The elements of (resp., ) are called inverse elements (resp., weak inverse elements) of . is called the sandwich set of and . We now list some results about -semigroups and -semigroups in the following results:
Result 1. (Result 2 [2]). Let be a semigroup and . Then, the following statements are equivalent.(1) is a -semigroup(2) is a regular subsemigroup of (3) .
Result 2. (Theorem 3.1 [3]). Let be a semigroup and . Then, the following statements are equivalent.(1) is an -semigroup(2) (3) (4) .
Result 3. (Proposition 3.4 [3]). Let be an -semigroup. Then,(1) (2) .A relationship between -semigroups and -semigroups established by congruences can be seen in [4]. From Results 1 and 2, we know that an -semigroup is a -semigroup. However, the converse is not true in general (for example, a regular (not orthodox) semigroup is a -semigroup, but it is not an -semigroup). In this note, we introduce the concept of -semigroups and give the conclusion that a semigroup is an -semigroup if and only if is a -semigroup and a -semigroup.
2. Main Results
Let be a semigroup. For and , we denote
It is clear that for any .
Definition 1. Let be a semigroup and . is called a -semigroup if is a subsemigroup for all , i.e., .
Proposition 1. Let be a -semigroup. Then, for any .
Proof. Let . Then, . Thus,Hence, .
Theorem 1. Let be a semigroup and . Then, is a -semigroup if and only if for all .
Proof. Let and . Then, there exists such that . It follows from Proposition 1 that . Also, . Thus, . Moreover, . Hence,Therefore, . From , we obtain the result.
Let and . Then, and . Thus, . It follows thatHence, . Therefore, we complete this proof.
Lemma 1. Let be a semigroup and . Then, the following two statements are equivalent.(1) for any (2) for any
Proof. (1) (2) Let for some . Then, . It follows thatThus, and . By condition (1), we have .
(2) (1) It follows from for any .
Theorem 2. Let be a semigroup and . Then, is a -semigroup if and only if the following two conditions hold.(1) or (1′) (2).
Proof. It follows from Proposition 1 that condition (1) holds. Next, we prove condition (2). Let . Then, we have for any . From Theorem 1, we have . Thus, . Hence, . By symmetry, .
Let . Then, there exists such that . Also, . Thus, and by condition (1). By condition (2), we have . Hence, . Therefore, from , and so, is a -semigroup from Theorem 1.
From Result 3 and Theorem 2, we obtain the following conclusion.
Proposition 2. Let be an -semigroup. Then, is a -semigroup.
In general, the converse is not true. See the following.
Example 1. (1) Let with the following operation:
We can see that . Moreover, and . Thus, is a -semigroup. However, is not a subsemigroup and so is not a -semigroup. Indeed, . Also, is not a -semigroup.
(2) Let be a group, andLet , and define an operation on byThen, is a regular semigroup and so is a -semigroup. It is easy to show that . However, . Hence, is not an -semigroup. Moreover, we can verify that . Also, . Thus, is not a -semigroup.
Next, we give the relationship between the three classes of semigroups.
Lemma 2. Let be a -semigroup and . Then, for any .
Proof. Let . Since is a -semigroup, . Let and . Then,Thus, . Moreover,and so, . Finally, it is clear that . Therefore, .
Theorem 3. Let be a semigroup and . Then, is an -semigroup if and only if the following two conditions hold.(1) is a -semigroup(2) or (2′) .
Proof. It follows from Results 1, 2, and 3.
Let . Since is a -semigroup, from Lemma 2, we have that there exists such that . By hypothesis, . It follows that is an -semigroup.
Finally, from Theorem 2, Proposition 2, and Theorem 3, we establish the relationship between -semigroups, -semigroups, and -semigroups by the following result.
Theorem 4. Let be a semigroup and . Then, is an -semigroup if and only if is a -semigroup and a -semigroup.
Remark 1. Let be a regular semigroup. Then, is a -semigroup. From Theorem 4, we know that is orthodox if and only if is a -semigroup (see Theorem 3.2 [5]).
Data Availability
The data used to support the findings of this study are available upon request to the corresponding author.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (No. 11701504), the Guangdong Basic and Applied Basic Research Foundation (No. 2022A1515011081), the Characteristic Innovation Project of Department of Education of Guangdong Province (No. 2020KTSCX159), the Science and Technology Innovation Guidance Project of Zhaoqing City (No. 2021040315026), the Innovative Research Team Project of Zhaoqing University, and the Scientific Research Ability Enhancement Program for Excellent Young Teachers of Zhaoqing University.