Abstract
In this paper, we determine necessary and sufficient conditions for BruckReilly and generalized BruckReilly ∗extensions of arbitrary monoids to be regular, coregular and stronglyπinverse. These semigroup classes have applications in various field of mathematics, such as matrix theory, discrete mathematics and padic analysis (especially in operator theory). In addition, while regularity and coregularity have so many applications in the meaning of boundaries (again in operator theory), inverse monoids and BruckReilly extensions contain a mixture fixedpoint results of algebra, topology and geometry within the purposes of this journal.
MSC: 20E22, 20M15, 20M18.
Keywords:
BruckReilly extension; generalized BruckReilly ∗extension; πinverse monoid; regular monoid1 Introduction and preliminaries
In combinatorial group and semigroup theory, for a finitely generated semigroup (monoid), a fundamental question is to find its presentation with respect to some (irreducible) system of generators and relators, and then classify it with respect to semigroup classes. In this sense, in [1], the authors obtained a presentation for the BruckReilly extension, which was studied previously by Bruck [2], Munn [3] and Reilly [4]. In different manners, this extension has been considered as a fundamental construction in the theory of semigroups. In detail, many classes of regular semigroups are characterized by BruckReilly extensions; for instance, any bisimple regular wsemigroup is isomorphic to a Reilly extension of a group [4] and any simple regular wsemigroup is isomorphic to a BruckReilly extension of a finite chain of groups [5,6]. After that, in another important paper [7], the author obtained a new monoid, namely the generalized BruckReilly ∗extension, and presented the structure of the ∗bisimple type Awsemigroup. Later on, in [8], the authors studied the structure theorem of the ∗bisimple type Asemigroups as the generalized BruckReilly ∗extension. Moreover, in a joint work [9], it has been recently defined a presentation for the generalized BruckReilly ∗extension and then obtained a GröbnerShirshov basis of this new construction. As we depicted in the abstract of this paper, BruckReilly, its general version generalized BruckReilly ∗extension of monoids and semigroup classes are not only important in combinatorial algebra but also in linear algebra, discrete mathematics and topology. So these semigroup classes, regular, coregular, inverse and strongly πinverse, are the most studied classes in algebra.
In this paper, as a next step of these above results, we investigate regularity, coregularity and stronglyπinverse properties over BruckReilly and generalized BruckReilly ∗extensions of monoids. We recall that regularity and strongly πinverse properties have been already studied for some other special extensions (semidirect and wreath products) of monoids [10,11]. We further recall that these two important properties have been also investigated for the semidirect product version of Schützenberger products of any two monoids [12,13]. However, there are not yet such investigations concerning coregularity. As we depicted in the abstract, semigroup classes have important applications in various fields of mathematics, such as matrix theory, discrete mathematics and padic analysis (especially in operator theory). In addition, while regularity and coregularity have so many applications in the meaning of boundaries (again in operator theory), inverse monoids and BruckReilly extensions contain a mixture of algebra, topology and geometry within the purposes of this journal.
Now let us present the following fundamental material that will be needed in this paper. We refer the reader to [1416] for more detailed knowledge.
An element a of a semigroup S is called regular if there exists such that . The semigroup S is called regular if all its elements are regular. Groups are of course regular semigroups, but the class of regular semigroups is vastly more extensive than the class of groups (see [16]). Further, to have an inverse element can also be important in a semigroup. Therefore, we call S is an inverse semigroup if every element has exactly one inverse. The wellknown examples of inverse semigroups are groups and semilattices. An element is called coregular and b its coinverse if . A semigroup S is said to be coregular if each element of S is coregular [17]. In addition, let and RegS be the set of idempotent and regular elements, respectively. We then say that S is called πregular if, for every , there is an such that . Moreover, if S is πregular and the set is a commutative subsemigroup of S, then S is called stronglyπinverse semigroup[16]. We recall that RegS is an inverse subsemigroup of a strongly πinverse semigroup S.
2 BruckReilly extensions of monoids
Let us suppose that A is a monoid with an endomorphism θ defined on it such that Aθ is in the ℋclass [16] of the identity of A. Also, let denotes the set of nonnegative integers. Hence, the set with the multiplication
where and is the identity map on A, forms a monoid with identity . Then this monoid is called the BruckReilly extension of A determined by θ[24] and denoted by .
In the above references, the authors used to prove that every semigroup embeds in a simple monoid, and to characterize special classes of inverse semigroups. In [[3], Theorem 3.1], Munn showed that is an inverse semigroup if and only if A is inverse. So, the following result is a direct consequence of this theorem.
Corollary 1LetAbe an arbitrary monoid. Thenis regular if and only ifAis regular.
with and , the set becomes a subsemigroup of . Thus, we further have the following lemma.
Lemma 1Let. Ifis coregular then.
Proof Let . Then there exists such that
We have
for some , where and . This implies and , in other words . Further, for some , we have
where and . This gives , , and consequently, . Together with , we obtain as required. □
Lemma 1 shows that a coregular element in and its coinverse belongs to
Now we can present the following result.
Theorem 1LetAbe a monoid. Thenis coregular if and only ifAis coregular.
Proof Assume that is a coregular monoid. For , there exists such that
and
By (1) and (2), we clearly have , and hence, and . So A is coregular.
Conversely, let . Then there is an with and . Thus, for , we get
and
In [[3], Theorem 3.1], it is proved that:
• is an idempotent element in if and only if and a is an idempotent element in A.
This result will be used in the proof of the following theorem.
Theorem 2is stronglyπinverse if and only ifAis regular and the idempotents inAcommute.
Proof Let be strongly πinverse, and let . Also let us consider the element in . Then there exists an element with . It is actually a routine matter to show that . Moreover, there exists an element such that is an inverse of (see [3]). Therefore,
This shows that
By the assumption given in the beginning of this section, since aθ is in the ℋclass of the element , we obtain aθ is a group element, and so there is an inverse element . Thus, by (3), we get ; in other words, . Consequently, A is regular. Now, let us also show that the elements in are commutative. But this is quite clear by the fact that the idempotents in commute if and only if the idempotents in A commute (see [[3], Theorem 3.1(5)]).
Conversely, let us suppose that A is regular and the idempotents in A commute. Then is regular, where πregular by Corollary 1. Moreover, again by [[3], Theorem 3.1(5)], is a commutative subsemigroup, which is required to satisfy strongly πinverse property, hence the result. □
3 The generalized BruckReilly ∗extension of monoids
Suppose that A is an arbitrary monoid having and as the  and ℋ classes containing the identity element of A. Moreover, let us assume that β and γ are morphisms from A into and, for an element u in , let be the inner automorphism of defined by such that .
Now one can consider the set into a semigroup with a multiplication
where and , are interpreted as the identity map of A, and also is interpreted as the identity of A. In [8], Yu Shung and LiMin Wang showed that S is a monoid with the identity . In fact, this new monoid is denoted by and called generalized BruckReilly ∗extension of A determined by the morphisms β, γ and the element u.
The following lemmas were established in [8].
Lemma 2If, thenis an idempotent if and only if, andvis idempotent.
if and only ifis an inverse ofvinAwhile, , and.
Then we have an immediate consequence as in the following.
Corollary 2LetAbe a monoid. Thenis regular if and only ifAis regular.
In this section, we mainly characterize the properties coregularity and strongly πinverse over the generalized BruckReilly ∗extensions of monoids. More specifically, for a given monoid A, we determine the maximal submonoid of , which can be held coregularity if A satisfies particular properties.
Our first observation is the following.
Lemma 4The setis a submonoid of.
Proof By considering the multiplication in (4), the proof can be seen easily. □
It turns out that all coregular elements in belong to the submonoid ℒ .
Lemma 5Let. Ifis coregular thenand.
Proof Let be a coregular element. Then there exists an element such that
for some , where and . This implies that
and
By (6), we have . Applying this in (5), we get and thus . Further, we have
for some , where and . This implies that , ,
and
By writing the equality (8) in (7), we get . Together with , we obtain . By assuming , we also get .
Now let . Then it is easy to verify that and . If , we can easily see that or . This shows that if or . Hence, or is not possible. □
Then we have the following result.
Theorem 3LetAbe a monoid. Then the submonoid ℒ ofis coregular if and only ifAis coregular.
Proof Suppose that is coregular. For each in ℒ, there exists an element such that
and
By (9) and (10), we obtain , and hence and . So, A is coregular.
Conversely, let A be a coregular monoid and . Then there exists an element with and . Therefore, for , we get
Hence, is a coregular monoid, as desired. □
In the final theorem, we consider strongly πinverse property.
Theorem 4is stronglyπinverse if and only ifAis regular and the idempotents inAcommute.
Proof We will follow the same format as in the proof of Theorem 2. So, let us suppose that is a strongly πinverse monoid, and let . Then, for , there is an element with . It is easily seen that . Moreover, there is an element such that is an inverse of by Lemma 3. From here, we have
This actually shows that
At the same time, since aβ is in the class of the , there exists an inverse element . Thus, by (11), we get , in other words, . Hence, A is regular. Now, let us show that the elements in are commutative to conclude the necessity part of the proof. To do that, consider any two elements and in . Thus, (by Lemma 2) and we have
Conversely, let us suppose that A is regular. Then is regular, where πregular by Corollary 2. Now we need to show that the elements in commute. To do that, let us take , and thus by Lemma 2. Now, by considering the multiplication as defined in (4), we have the following cases.
and
respectively, where . Since , we deduce that both and are the elements of , in other words,
Case (ii): If or , then we get
or
respectively. Since , we clearly obtain .
Hence, the result. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors completed the paper together. All authors read and approved the final manuscript.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
The second and fourth authors are partially supported by Research Project Offices (BAP) of Selcuk (with Project No. 13701071) and Uludag (with Project No. 201215 and 201219) Universities, respectively.
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