In this paper, we determine necessary and sufficient conditions for Bruck-Reilly and generalized Bruck-Reilly ∗-extensions of arbitrary monoids to be regular, coregular and stronglyπ-inverse. These semigroup classes have applications in various field of mathematics, such as matrix theory, discrete mathematics and p-adic 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 Bruck-Reilly extensions contain a mixture fixed-point results of algebra, topology and geometry within the purposes of this journal.
MSC: 20E22, 20M15, 20M18.
Keywords:Bruck-Reilly extension; generalized Bruck-Reilly ∗-extension; π-inverse monoid; regular monoid
1 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 , the authors obtained a presentation for the Bruck-Reilly extension, which was studied previously by Bruck , Munn  and Reilly . 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 Bruck-Reilly extensions; for instance, any bisimple regular w-semigroup is isomorphic to a Reilly extension of a group  and any simple regular w-semigroup is isomorphic to a Bruck-Reilly extension of a finite chain of groups [5,6]. After that, in another important paper , the author obtained a new monoid, namely the generalized Bruck-Reilly ∗-extension, and presented the structure of the ∗-bisimple type Aw-semigroup. Later on, in , the authors studied the structure theorem of the ∗-bisimple type A-semigroups as the generalized Bruck-Reilly ∗-extension. Moreover, in a joint work , it has been recently defined a presentation for the generalized Bruck-Reilly ∗-extension and then obtained a Gröbner-Shirshov basis of this new construction. As we depicted in the abstract of this paper, Bruck-Reilly, its general version generalized Bruck-Reilly ∗-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 π-inverse, are the most studied classes in algebra.
In this paper, as a next step of these above results, we investigate regularity, coregularity and stronglyπ-inverse properties over Bruck-Reilly and generalized Bruck-Reilly ∗-extensions of monoids. We recall that regularity and strongly π-inverse 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 p-adic 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 Bruck-Reilly extensions contain a mixture of algebra, topology and geometry within the purposes of this journal.
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 ). 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 well-known 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 . In addition, let and RegS be the set of idempotent and regular elements, respectively. We then say that S is called π-regular if, for every , there is an such that . Moreover, if S is π-regular and the set is a commutative subsemigroup of S, then S is called stronglyπ-inverse semigroup. We recall that RegS is an inverse subsemigroup of a strongly π-inverse semigroup S.
2 Bruck-Reilly extensions of monoids
Let us suppose that A is a monoid with an endomorphism θ defined on it such that Aθ is in the ℋ-class  of the identity of A. Also, let denotes the set of nonnegative integers. Hence, the set with the multiplication
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 [, 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.
Now we can present the following result.
In [, Theorem 3.1], it is proved that:
This result will be used in the proof of the following theorem.
Proof Let be strongly π-inverse, 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 ). Therefore,
This shows that
By the assumption given in the beginning of this section, since aθ is in the ℋ-class 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 [, Theorem 3.1(5)]).
Conversely, let us suppose that A is regular and the idempotents in A commute. Then is regular, where π-regular by Corollary 1. Moreover, again by [, Theorem 3.1(5)], is a commutative subsemigroup, which is required to satisfy strongly π-inverse property, hence the result. □
3 The generalized Bruck-Reilly ∗-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 .
where and , are interpreted as the identity map of A, and also is interpreted as the identity of A. In , Yu Shung and Li-Min Wang showed that S is a monoid with the identity . In fact, this new monoid is denoted by and called generalized Bruck-Reilly ∗-extension of A determined by the morphisms β, γ and the element u.
The following lemmas were established in .
Then we have an immediate consequence as in the following.
In this section, we mainly characterize the properties coregularity and strongly π-inverse over the generalized Bruck-Reilly ∗-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.
Proof By considering the multiplication in (4), the proof can be seen easily. □
Then we have the following result.
In the final theorem, we consider strongly π-inverse property.
Proof We will follow the same format as in the proof of Theorem 2. So, let us suppose that is a strongly π-inverse 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 π-regular 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.
Hence, the result. □
The authors declare that they have no competing interests.
All authors completed the paper together. All authors read and approved the final manuscript.
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. 2012-15 and 2012-19) Universities, respectively.
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