Abstract
In this study, we define new semigroup structures using the set which is called the source of semiprimeness for a semigroup with zero element. idempotent semigroup, regular semigroup, reduced semigroup, and nonzero divisor semigroup which are generalizations of idempotent, regular, reduced, and nonzero divisor semigroups in semigroup theory are investigated, and their basic properties are determined. In addition, we adapt some well-known results in semigroup theory to these new semigroups.
1. Introduction
Semiprime ideals play a very important role in semigroups. Since every ring can be considered as a semigroup under multiplication, we have more generalized theorems of [1]. The aim of this study is to obtain new semigroup structures by using the definition of semiprimeness in semigroups in the sense of the study by Aydın et al. [1]. There are different equivalent definitions of semiprimeness. One of these is, if with implies , then is called a semiprime semigroup. In [1], the authors have defined the set called source of semiprimeness of ring by using the ring version of semiprimeness definition. In addition, they have obtained new ring and field structures. In this study, new semigroup structures have been obtained by using the source set of semiprimeness of semigroups. These structures are investigated, and some results of the semigroup theory are adapted to the new semigroups.
2. Preliminaries
In semigroup theory, there are many studies on different semigroup types. Adams examined the properties of semiprime semigroups in [2]. In [3], Van Rooyen worked on the ideals of semigroups. In [4–6], the authors have worked on reduced, regular, and zero-divisor semigroups. This article will bring a different perspective to these various semigroups.
First, let us give the definitions mentioned in this article which are frequently used in semigroup theory. The studies by Clifford and Preston and Grillet [7, 8] are used for definitions.
Let be a semigroup. A zero element of a semigroup is an element , such that for all . Throughout this study, will be taken as a semigroup with zero element. The element, such that , is called an idempotent element. The semigroup whose all elements are idempotent is called an idempotent semigroup. An element of a semigroup is called regular element if there exist at least one , such that . The semigroup whose all elements are regular is called a regular semigroup. For , if there exist , such that and , then is called nilpotent. A semigroup without nilpotent elements other than zero is called a reduced semigroup. An element of a semigroup is called zero divisor element if there exist , such that (left zero divisor) and (right zero divisor). S is called the nonzero divisor semigroup, if there is no nonzero zero divisor element in .
Now, let us give the definitions of ideal and semiprime semigroup which are the basis of our article. From [2, 9], the subset of semigroup is called an ideal if (right ideal) and (left ideal). One of the equivalent definition of the semiprime semigroup, which using this study, is given as follows: if with implies , then is called the semiprime semigroup.
3. Results of Different Types of Semigroups
Definition 1. Let be a semigroup with zero and be a nonempty subset of . The subset of ,is called the source of semiprimeness of in . For semigroup , the notation is used instead of . Then, the source of semiprimeness of the semigroup is defined as follows:
Now, let us examine the basic properties of the source of semiprimeness of semigroup . In [1], the set of source of semiprimeness was investigated for the rings. The properties that can be exactly provided for semigroups are given with reference without proof in this study. First of all, let us mention some facts that can be easily seen but useful for understanding the set.(1)For every semigroup with zero, since , is provided for every subset of (2)It is easy to see that for the subsemigroup of (3)For subsets and , implies ([1], Proposition 2.2)
Now, let us investigate the properties of idempotent, regular, nilpotent, and zero divisor elements. These properties form the basis for defining the new semigroup structures that we will define in the next section. By using the results obtained, we will obtain the definitions of the idempotent semigroup, regular semigroup, reduced semigroup, and nonzero divisor semigroup.
Lemma 1. Let be a semigroup, be an idempotent element, and be a regular element, such that for . Then, the following properties are provided.(1)(2)(3)If is an idempotent semigroup, then (4)If is a regular semigroup, then (5)If , then is a nilpotent element(6)If , then is a zero divisor element
Proof. (1)Let . Then, since , we write . From this equation, we get This gives us . So, is provided.(2)For regular element , such that for , this implies that , and is an idempotent element. From (1), it is clear that .(3)Let be an idempotent semigroup and . Since , the equation is satisfied for . Hence, using as an idempotent element, we get . Then, .(4)Let be a regular semigroup and . In this case, , and there exist , such that . Specially, is provided. This means that .(5)If , then . Since , is a nilpotent element.(6)If , then . This equation can be written as and . Since , if , then is a zero divisor element. On the other hand, if , then is also a zero divisor element specific with .
Using the above Lemma 1, it is easy to see that the following corollary.
Corollary 1. For semigroup , the following holds true:(1)There is no idempotent element other than zero in (2)There is no regular element other than zero in (3)Every element in is nilpotent element(4)Every element in is zero divisor element
Now, we will define different type of semigroups.
Definition 2. Let be semigroup with zero and .(1) is called the idempotent semigroup if every element of is idempotent(2) is called the regular semigroup if every element of is regular(3) is called the reduced semigroup if has no nilpotent element(4) is called the nonzero divisor semigroup if has neither a left nor a right zero divisor
First, let us mention the basic characteristics properties of these newly defined semigroups. The following results can be easily seen using the definitions.(1)If , then . In this case, since , definitions would be meaningless for the zero semigroup. Similarly, is provided in case of . So, in this case too, the definitions would be meaningless.(2)For the elements of a semigroup, “if is an idempotent element, then is regular element,” “if is a nilpotent element, then is a zero divisor element,” and “if is a nilpotent element, then is not an idempotent element” and conditions are always provided. Using these conditions, the following properties can be easily seen. Additional conditions for providing other directions of relations will be investigated in our study(3)It is clear that, if is an idempotent (regular, reduced, and nonzero divisor) semigroup, then is a idempotent (regular, reduced, and nonzero divisor) semigroup
We will give examples of each of these four semigroups before proceeding with conclusions. It can also be seen from these examples that the above relations are one sided.
Example 1. Let the operation table of the semigroup be given as follows.Using the table, we getSince , it is seen that is idempotent and regular element of . So, is a idempotent and regular semigroup. Also, since is not nilpotent, is a reduced semigroup. However, is a zero divisor element for and . Thus, is not a nonzero divisor semigroup.
Example 2. Let the operation table of the semigroup be given as follows.For the semigroup , it is easy to see thatIf we investigate elements of , we see that only is an idempotent and regular element. So, is not a idempotent or regular semigroup. Also, since and , and are the zero divisor elements. Then, is not a nonzero divisor semigroup. On the other hand, it is clear that and are not nilpotent elements. So, is a reduced semigroup.
Example 3. Let the operation table of the semigroup be given as follows.Now, it turns out thatIt is seen that and are the idempotent and regular elements of . Then, is a idempotent and a regular semigroup. Also, and are nonzero divisors. Therefore, is a nonzero divisor semigroup. On the other hand, since and are not nilpotent elements, is also a reduced semigroup.
Example 4. Let the operation table of the semigroup be given as follows.Using the table, we getIf we investigate the elements of , we see that only is an idempotent element. Then, is not a idempotent semigroup. However, since and , and are the regular elements. So, is a regular semigroup. Also, and are not zero divisor or nilpotent elements. Therefore, is a nonzero divisor and a reduced semigroup.
Example 5. Consider the semigroup . Sincewe getIn the set , since only 1 is idempotent and regular, is not a idempotent or regular semigroup. On the other hand, since there are no zero divisor elements in , S is a nonzero divisor and reduced semigroup.
Let us now give a proposition and an example about characterization of the subgroups of these new semigroups.
Proposition 1. Let be a semigroup and be a subsemigroup of . Then, the following conditions are satisfied.(1)If is a idempotent (regular) semigroup, then is a idempotent (regular) semigroup(2)If is a reduced (nonzero divisor) semigroup, then is a reduced (nonzero divisor) semigroup
Proof. (1)If , then and . Thus, we write and . This means that . So, . Therefore, since every element in is idempotent (regular), every element in is an idempotent (regular) element. So, is a idempotent (regular) semigroup.(2)We showed that in . So, since there is no nilpotent (zero divisor) element in , there is no nilpotent (zero divisor) element in . In this case, is a reduced (nonzero divisor) semigroup.
Example 6. Consider the set is a semigroup with zero element by multiplication operation in matrices. For this semigroup, it is not hard to see thatSince there exist , such that for each , is a regular semigroup. On the other hand, the elements of the set can only be nilpotent elements for . So, is a reduced semigroup. Also, it is easy to see that is a nonzero divisor semigroup because there is no zero divisor element in .
Now, let define the subsemigroup,of semigroup . From Proposition 1, is a regular, reduced, and nonzero divisor semigroup. Indeed, it is clear thatIf we investigate similar to the above, we see that all elements are regular elements, and there are no nilpotent or zero divisor elements.
Obviously, the set does not have to be a subsemigroup. The following proposition shows that the set is the subsemigroup with additional conditions for semigroup types. In the next Lemma, the characterization of the is given for the nonzero divisor and reduced semigroups.
Proposition 2. Let be a semigroup. Then, the following conditions are satisfied.(1)If is a nonzero divisor semigroup, then is a subsemigroup(2)If is a commutative and idempotent (regular) semigroup, then is a subsemigroup
Proof. (1)Let be a nonzero divisor semigroup. If , then and are the nonzero divisors. Then, element is also a nonzero divisor. This gives us . So, is a subsemigroup.(2)Note that product of idempotent elements is an idempotent element, product of regular elements is a regular element, and product of nilpotent elements is a nilpotent element in a commutative semigroup. In this case, if , then for the idempotent (regular) semigroup . So, is a subsemigroup.
Lemma 2. If is a nonzero divisor or reduced semigroup, then . In addition, if is a monoid, then is provided for nonzero divisor or reduced semigroup .
Proof. First, let be a nonzero divisor semigroup and . The inclusion is always provided. Conversely, let us take an arbitrary element of . Assume that . Since is a nonzero divisor semigroup, is a nonzero divisor element. From definition of the set , we write . Using as a nonzero divisor, we get . Similarly, from the equation , we get . However, this result leads us to the contradiction. So the assumption is incorrect, and thus, . Then, is provided.
Now, let be a reduced semigroup. Similar to above, is always provided. Conversely, if , then . This gives us is a nilpotent element. Since is a reduced semigroup, there is no nilpotent element in . So, we get . Hence is provided.
In addition, if is a monoid, then there exist , such that for all . This means that . Also, similar to the above proof, we obtain .
In this part of our study, we will give the results about is an semigroup. As it is known, if monoid has , such that for , is called an unit element. Since is a monoid with inverse and unit elements, different results can be reached from other semigroups. In [1], studies on domains and reduced rings were given. In the following part, similar results are obtained by different methods for semigroups.
Lemma 3. Let be semigroup of integers modulo . If is for prime number , then . In addition, for .
Proof. Let for prime number . From Lemma 2, we write for an arbitrary element of . This means that . Since is prime, we get , and so, we obtain . Then, for . So, we get .
Let be a prime number and of . Using the same procedure as in the above paragraph, we have . Since is a prime number, the only element that can be divided by is itself. So, we get . This result gives us . So, we get .
Theorem 1. Let be semigroup of integers modulo . The following holds true:(1)For , is not a idempotent monoid(2)If is either a prime or , then is a regular monoid(3) is either a prime or if and only if is a nonzero divisor monoid(4)If is a prime number, then is a reduced monoid
Proof. (1)Since , we get . It is obvious that is an idempotent element. Therefore, is a idempotent monoid. Now, we consider monoid for . Let be an unit element of . Then, for all . Specially, is provided for . This result leads us to the contradiction . This means that there is no unit element in . So, every unit element is element of . In this case, if is an unit element in , is also an idempotent element. However, an unit element different from the identity cannot be an idempotent element. Therefore, we get . So, the assumption is incorrect, and thus, is not idempotent monoid.(2)We recall that if , then is an unit element. Let be a prime number. Then, we get from Lemma 3. In this case, every element in is an unit element. Since an unit element is also a regular element, is a regular monoid. Let for prime number . Then, we have from Lemma 3. This means that, if , then . So, is an unit element. Since an unit element is also a regular element, is a regular monoid.(3)Let be either a prime or . We showed that in , if , then is an unit element. It is well-known that, an unit element cannot be a zero divisor in . Therefore, there is no zero divisor in . So, is a nonzero divisor monoid. Conversely, let be a nonzero divisor monoid. In this case, every element in is a nonzero divisor. This means that is an unit element, and . These results provide for any prime number. Assume that is not prime and for prime number and for some . Since , is a nonunit element. Hence, must be in , and so, from Lemma 2. Therefore, we get . Since , we write . This means that , and so . Since , we obtain . So, we get .(4)Let be a prime number. We showed that in , if , then is an unit element. If , then . If we continue in a similar way, then we obtain contradiction. Thus, there is no nilpotent element in . This means that is a reduced monoid.
Example 7. Consider the monoid . For this monoid, we getFrom Theorem 1, is not a idempotent monoid. Besides that , is not regular and nonzero divisor monoid. Indeed, if the elements of are investigated, it is seen that is not an idempotent element. However, since and , these two elements are regular elements. Then, is a regular monoid. Also, since and are not zero divisors or nilpotent elements, is a nonzero divisor and reduced monoid.
Example 8. Consider the monoid . Now, it turns out thatFrom Theorem 1, is not a idempotent monoid. Indeed, only element is idempotent in . Also, since is not regular element, is not a regular monoid. On the other hand, since is zero divisor and nilpotent, is a not nonzero divisor or reduced monoid.
In [10], the relation between the regular semigroups and their ideals was investigated and characterization of the regular semigroups was given. We will now investigate the relation of regular semigroups with their ideals.
Lemma 4. Let and are the nonempty subsemigroups of semigroup . Then, the following properties are provided.(1)(2)If is left (right, both sided) ideal, then is right (left, both sided) ideal(3)If is ideal, then is ideal. Specifically, is ideal of .(4)If and are ideals, then (5)If and are ideals, then
Proof. We will prove (1), (2), and (4). (3) and (5) are easily seen; similarly,(1)If , then . Since , we get . So, .(2)Let be a left ideal and . Then, we write and . For arbitrary , since is a left ideal, we have . So, the equation is satisfied. This means that , and so, is right ideal.(4)Let . Then, we get and . Since is an ideal, we obtain . So, we get This result means that . So, is provided.
Theorem 2. Let be a commutative semigroup. is a regular semigroup if and only if for every ideals and .
Proof. Let be a regular semigroup and . Thus, and . Assume that . From Lemma 1 and Proposition 2, we get . This result contradicts with . Then, and . This gives us as a regular element. Therefore, there exists , such that . In this equation, since and are ideals, we write . Also, since is a regular element and using Proposition 1, is provided. Then, we get . From this expression, we obtainOn the other hand, if , then is a regular element because and . Also, since , we have . In this expression, using for regular element , we obtain . Therefore,is provided. So, we get .
To prove the converse, let for right ideal and left ideal , and let . Let us consider the ideal by generated . From the hypothesis, we getUsing and , we obtain . Therefore, we have . Also, we know that for commutative semigroup . From the hypothesis, we getThis equation gives us . This means that is a regular element such that for some . So, is a regular semigroup.
Theorem 3. Let be a commutative semigroup. is a regular semigroup if and only if is provided for each ideal of .
Proof. Let be a regular semigroup. For ideal , using Theorem 2, we getConversely, let for any ideal . Then, since is an ideal, we getIn above equation, using from Lemma 4 (5) and , we obtainSo, using , we getAlso, since ,is provided. Then, . From Theorem 2, is a regular semigroup.
Now, let us give the relation between regular and reduced semigroups. The following example shows that the commutative property in this theorem is necessary.
Theorem 4. If is a commutative regular semigroup, then is a reduced semigroup.
Proof. Let be a commutative regular semigroup, and let be a nilpotent element. In this case, there exists , such that and . is also a regular element. Since is commutative, there exists , such that . Thus, we getHowever, this result contradicts . Therefore, there is no nilpotent element in . So, is a reduced semigroup.
Example 9. Let the operation table of the semigroup be given as follows.Using the table, we getSince is not an idempotent element, is not a idempotent semigroup. On the other hand, since , , and , every element in is regular. So, is a regular semigroup. However, is not a reduced semigroup because and are the nilpotent elements. Therefore, is not a nonzero divisor either. As shown in this example, in a noncommutative semigroup, the above result is not valid.
Finally, we will adapt one of the well-known results in semigroup theory to regular semigroups. An element of a semigroup S is called inverse element if there exist at least one , such that and . If there is an inverse element with uniqueness for each element of the semigroup, the semigroup is called the inverse semigroup. As can be seen in [8], Proposition 2.6, is an inverse semigroup if and only if is a regular and the idempotents of commute with each other. We reach similar results for regular semigroups, and a relation establishes between regular semigroups and inverse semigroups.
Lemma 5. Every elements of are inverse if and only if is a regular semigroup.
Proof. If every elements of are inverse, then these elements are also regular elements. It is clear that is a regular semigroup.
Conversely, let be a regular semigroup. Then, each element is regular, and there exists , such that .So, and are inverse for each other. So, every elements in are inverse elements.
Theorem 5. Every elements of are inverse elements with uniqueness if and only if is a regular semigroup and the idempotents of commute with each other.
Proof. From Lemma 5, every elements of are inverse if and only if is a regular semigroup. As can be seen in [8], Proposition 2.6, is an inverse semigroup if and only if is a regular and the idempotents of commute with each other. From this property, all idempotent elements of commute with each other.
4. Conclusions
We have shown that some properties of the source of the semiprimeness defined as for a semigroup are given. Moreover, the relations of the source of the semiprimeness with idempotent, regular, nilpotent, and zero divisor elements, which are the basis of the new semigroup structures, are investigated. Additionally, we define the idempotent semigroup, regular semigroup, reduced semigroup, and nonzero divisor semigroup structures. Thus, the mentioned semigroups are generalized. Also, we give examples for each semigroup. In particular, the monoid is investigated and generalizations are obtained. Furthermore, we adapt some well-known results in semigroup theory to new semigroup structures. The source of primeness can be investigated in the sense of this article in future works.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Disclosure
This manuscript is an extended and improved version of the abstract named “A Note on the Source of Semiprimeness of Semigroups” presented in the 9th Mas International European Conference on Mathematics.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by Çanakkale Onsekiz Mart University, the Scientific Research Coordination Unit (project number: FBA-2019-2792).