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This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Some fixed-point results on (generalized) Bruck-Reilly ∗-extensions of monoids

Eylem Guzel Karpuz1*, Ahmet Sinan Çevik2, Jörg Koppitz3 and Ismail Naci Cangul4

Author Affiliations

1 Department of Mathematics, Kamil Özdag Science Faculty, Karamanoglu Mehmetbey University, Yunus Emre Campus, Karaman, 70100, Turkey

2 Department of Mathematics, Faculty of Science, Selçuk University, Campus, Konya, 42075, Turkey

3 Institute of Mathematics, Potsdam University, Potsdam, 14469, Germany

4 Department of Mathematics, Faculty of Arts and Science, Uludag University, Gorukle Campus, Bursa, 16059, Turkey

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Fixed Point Theory and Applications 2013, 2013:78  doi:10.1186/1687-1812-2013-78


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/78


Received:28 January 2013
Accepted:21 March 2013
Published:29 March 2013

© 2013 Guzel Karpuz et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 [1], the authors obtained a presentation for the Bruck-Reilly 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 Bruck-Reilly extensions; for instance, any bisimple regular w-semigroup is isomorphic to a Reilly extension of a group [4] 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 [7], 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 [8], the authors studied the structure theorem of the ∗-bisimple type A<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M1">View MathML</a>-semigroups as the generalized Bruck-Reilly ∗-extension. Moreover, in a joint work [9], 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.

Now let us present the following fundamental material that will be needed in this paper. We refer the reader to [14-16] for more detailed knowledge.

An element a of a semigroup S is called regular if there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M2">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M3">View MathML</a>. 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 well-known examples of inverse semigroups are groups and semilattices. An element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M4">View MathML</a> is called coregular and b its coinverse if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M5">View MathML</a>. A semigroup S is said to be coregular if each element of S is coregular [17]. In addition, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M6">View MathML</a> and RegS be the set of idempotent and regular elements, respectively. We then say that S is called π-regular if, for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M7">View MathML</a>, there is an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M8">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M9">View MathML</a>. Moreover, if S is π-regular and the set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M6">View MathML</a> is a commutative subsemigroup of S, then S is called stronglyπ-inverse semigroup[16]. 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 is in the ℋ-class [16] of the identity <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M11">View MathML</a> of A. Also, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M12">View MathML</a> denotes the set of nonnegative integers. Hence, the set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M13">View MathML</a> with the multiplication

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M14">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M15">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M16">View MathML</a> is the identity map on A, forms a monoid with identity <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M17">View MathML</a>. Then this monoid is called the Bruck-Reilly extension of A determined by θ[2-4] and denoted by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18">View MathML</a>.

In the above references, the authors used <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18">View MathML</a> 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 <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18">View MathML</a> 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. Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18">View MathML</a>is regular if and only ifAis regular.

For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M22">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M23">View MathML</a>, since

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M24">View MathML</a>

with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M25">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M26">View MathML</a>, the set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M27">View MathML</a> becomes a subsemigroup of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18">View MathML</a>. Thus, we further have the following lemma.

Lemma 1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M29">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M30">View MathML</a>is coregular then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M31">View MathML</a>.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M29">View MathML</a>. Then there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M33">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M34">View MathML</a>

We have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M35">View MathML</a>

for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M36">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M37">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M38">View MathML</a>. This implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M39">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M40">View MathML</a>, in other words <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M41">View MathML</a>. Further, for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M42">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M43">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M44">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M45">View MathML</a>. This gives <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M46">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M47">View MathML</a>, and consequently, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M48">View MathML</a>. Together with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M41">View MathML</a>, we obtain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M31">View MathML</a> as required. □

Lemma 1 shows that a coregular element in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18">View MathML</a> and its coinverse belongs to

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M52">View MathML</a>

Now we can present the following result.

Theorem 1LetAbe a monoid. Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M53">View MathML</a>is coregular if and only ifAis coregular.

Proof Assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M54">View MathML</a> is a coregular monoid. For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M55">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M56">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M57">View MathML</a>

(1)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M58">View MathML</a>

(2)

By (1) and (2), we clearly have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M59">View MathML</a>, and hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M60">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M61">View MathML</a>. So A is coregular.

Conversely, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M62">View MathML</a>. Then there is an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M63">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M60">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M61">View MathML</a>. Thus, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M66">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M67">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M68">View MathML</a>

Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M53">View MathML</a> is coregular. □

In [[3], Theorem 3.1], it is proved that:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M30">View MathML</a> is an idempotent element in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M31">View MathML</a> and a is an idempotent element in A.

This result will be used in the proof of the following theorem.

Theorem 2<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18">View MathML</a>is stronglyπ-inverse if and only ifAis regular and the idempotents inAcommute.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18">View MathML</a> be strongly π-inverse, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M75">View MathML</a>. Also let us consider the element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M76">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18">View MathML</a>. Then there exists an element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M78">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M79">View MathML</a>. It is actually a routine matter to show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M80">View MathML</a>. Moreover, there exists an element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M63">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M82">View MathML</a> is an inverse of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M83">View MathML</a> (see [3]). Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M84">View MathML</a>

This shows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M85">View MathML</a>

(3)

By the assumption given in the beginning of this section, since is in the ℋ-class of the element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M11">View MathML</a>, we obtain is a group element, and so there is an inverse element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M87">View MathML</a>. Thus, by (3), we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M88">View MathML</a>; in other words, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M89">View MathML</a>. Consequently, A is regular. Now, let us also show that the elements in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M90">View MathML</a> are commutative. But this is quite clear by the fact that the idempotents in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18">View MathML</a> 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 <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18">View MathML</a> is regular, where π-regular by Corollary 1. Moreover, again by [[3], Theorem 3.1(5)], <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M93">View MathML</a> is a commutative subsemigroup, which is required to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M18">View MathML</a> satisfy strongly π-inverse property, hence the result. □

3 The generalized Bruck-Reilly ∗-extension of monoids

Suppose that A is an arbitrary monoid having <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M95">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M96">View MathML</a> as the <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M97">View MathML</a>- and ℋ- classes containing the identity element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M11">View MathML</a> of A. Moreover, let us assume that β and γ are morphisms from A into <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M95">View MathML</a> and, for an element u in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M96">View MathML</a>, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M101">View MathML</a> be the inner automorphism of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M95">View MathML</a> defined by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M103">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M104">View MathML</a>.

Now one can consider the set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M105">View MathML</a> into a semigroup with a multiplication

(4)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M107">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M108">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M109">View MathML</a> are interpreted as the identity map of A, and also <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M110">View MathML</a> is interpreted as the identity <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M11">View MathML</a> of A. In [8], Yu Shung and Li-Min Wang showed that S is a monoid with the identity <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M112">View MathML</a>. In fact, this new monoid <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M105">View MathML</a> is denoted by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M114">View MathML</a> and called generalized Bruck-Reilly ∗-extension of A determined by the morphisms β, γ and the element u.

The following lemmas were established in [8].

Lemma 2If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M115">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M116">View MathML</a>is an idempotent if and only if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M117">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M118">View MathML</a>andvis idempotent.

Lemma 3If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M115">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M116">View MathML</a>has an inverse

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M121">View MathML</a>

if and only if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M122">View MathML</a>is an inverse ofvinAwhile<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M123">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M124">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M125">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M126">View MathML</a>.

Then we have an immediate consequence as in the following.

Corollary 2LetAbe a monoid. Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M114">View MathML</a>is regular if and only ifAis regular.

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 <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M114">View MathML</a>, which can be held coregularity if A satisfies particular properties.

Our first observation is the following.

Lemma 4The set<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M129">View MathML</a>is a submonoid of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M114">View MathML</a>.

Proof By considering the multiplication in (4), the proof can be seen easily. □

It turns out that all coregular elements in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M114">View MathML</a> belong to the submonoid ℒ .

Lemma 5Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M115">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M116">View MathML</a>is coregular then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M117">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M118">View MathML</a>.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M115">View MathML</a> be a coregular element. Then there exists an element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M137">View MathML</a> such that

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M139">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M126">View MathML</a>. Then we have

for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M142">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M143">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M144">View MathML</a>. This implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M145">View MathML</a>

(5)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M146">View MathML</a>

(6)

By (6), we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M147">View MathML</a>. Applying this in (5), we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M148">View MathML</a> and thus <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M149">View MathML</a>. Further, we have

for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M151">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M152">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M153">View MathML</a>. This implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M154">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M155">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M156">View MathML</a>

(7)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M157">View MathML</a>

(8)

By writing the equality (8) in (7), we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M158">View MathML</a>. Together with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M149">View MathML</a>, we obtain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M118">View MathML</a>. By assuming <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M139">View MathML</a>, we also get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M117">View MathML</a>.

Now let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M163">View MathML</a>. Then it is easy to verify that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M164">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M165">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M166">View MathML</a>, we can easily see that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M167">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M168">View MathML</a>. This shows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M169">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M170">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M171">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M170">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M171">View MathML</a> is not possible. □

Then we have the following result.

Theorem 3LetAbe a monoid. Then the submonoidof<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M114">View MathML</a>is coregular if and only ifAis coregular.

Proof Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M175">View MathML</a> is coregular. For each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M176">View MathML</a> in ℒ, there exists an element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M177">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M178">View MathML</a>

(9)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M179">View MathML</a>

(10)

By (9) and (10), we obtain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M180">View MathML</a>, and hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M181">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M182">View MathML</a>. So, A is coregular.

Conversely, let A be a coregular monoid and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M183">View MathML</a>. Then there exists an element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M184">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M181">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M182">View MathML</a>. Therefore, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M187">View MathML</a>, we get

Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M189">View MathML</a> is a coregular monoid, as desired. □

In the final theorem, we consider strongly π-inverse property.

Theorem 4<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M114">View MathML</a>is stronglyπ-inverse 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 <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M114">View MathML</a> is a strongly π-inverse monoid, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M75">View MathML</a>. Then, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M193">View MathML</a>, there is an element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M78">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M195">View MathML</a>. It is easily seen that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M196">View MathML</a>. Moreover, there is an element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M63">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M198">View MathML</a> is an inverse of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M199">View MathML</a> by Lemma 3. From here, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M200">View MathML</a>

This actually shows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M201">View MathML</a>

(11)

At the same time, since is in the <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M97">View MathML</a>-class of the <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M11">View MathML</a>, there exists an inverse element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M204">View MathML</a>. Thus, by (11), we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M205">View MathML</a>, in other words, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M89">View MathML</a>. Hence, A is regular. Now, let us show that the elements in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M90">View MathML</a> are commutative to conclude the necessity part of the proof. To do that, consider any two elements <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M208">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M209">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M90">View MathML</a>. Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M211">View MathML</a> (by Lemma 2) and we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M212">View MathML</a>

So, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M213">View MathML</a>, as required.

Conversely, let us suppose that A is regular. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M114">View MathML</a> is regular, where π-regular by Corollary 2. Now we need to show that the elements in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M215">View MathML</a> commute. To do that, let us take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M216">View MathML</a>, and thus <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M217">View MathML</a> by Lemma 2. Now, by considering the multiplication <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M218">View MathML</a> as defined in (4), we have the following cases.

Case (i): If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M154">View MathML</a>, then we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M220">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M221">View MathML</a>

respectively, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M222">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M223">View MathML</a>, we deduce that both <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M224">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M225">View MathML</a> are the elements of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M90">View MathML</a>, in other words,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M227">View MathML</a>

Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M228">View MathML</a>.

Case (ii): If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M229">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M230">View MathML</a>, then we get

or

respectively. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M233">View MathML</a>, we clearly obtain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/78/mathml/M228">View MathML</a>.

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. 2012-15 and 2012-19) Universities, respectively.

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