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

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R E S E A R C H

Open Access

Some ﬁxed-point results on (generalized)

Bruck-Reilly

∗

-extensions of monoids

Eylem Guzel Karpuz

1*

_{, Ahmet Sinan Çevik}

2

_{, Jörg Koppitz}

3

_{and Ismail Naci Cangul}

4

*_{Correspondence:}

[email protected]

1_{Department of Mathematics, Kamil}

Özdag Science Faculty, Karamanoglu Mehmetbey University, Yunus Emre Campus, Karaman, 70100, Turkey Full list of author information is available at the end of the article

Abstract

In this paper, we determine necessary and suﬃcient conditions for Bruck-Reilly and generalized Bruck-Reilly∗-extensions of arbitrary monoids to beregular, coregularand strongly

π

-inverse. Thesesemigroup classeshave applications in various ﬁeld of mathematics, such as matrix theory, discrete mathematics andp-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 (irre-ducible) system of generators and relators, and then classify it with respect to semigroup classes. In this sense, in [], the authors obtained a presentation for theBruck-Reilly ex-tension,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 regularw-semigroup is isomorphic to a Reilly extension of a group [] and any simple regularw-semigroup is isomorphic to a Bruck-Reilly extension of a finite chain of groups [, ]. After that, in another important paper [], the author obtained a new monoid, namely thegeneralized Bruck-Reilly∗-extension, and presented the structure of the∗-bisimple typeA w-semigroup. Later on, in [], the authors studied the structure theorem of the∗-bisimple type A w_{-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 pa-per, 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 al-gebra, discrete mathematics and topology. So these semigroup classes, regular, coregular, inverse and stronglyπ-inverse, are the most studied classes in algebra.

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In this paper, as a next step of these above results, we investigate regularity, coregu-larity andstronglyπ-inverseproperties over Bruck-Reilly and generalized Bruck-Reilly

∗-extensions of monoids. We recall that regularity and stronglyπ-inverse properties have been already studied for some other special extensions (semidirect and wreath products) of monoids [, ]. We further recall that these two important properties have been also investigated for the semidirect product version of Schützenberger products of any two monoids [, ]. However, there are not yet such investigations concerning coregularity. As we depicted in the abstract,semigroup classeshave important applications in various ﬁelds of mathematics, such as matrix theory, discrete mathematics andp-adic analysis (es-pecially 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 [–] for more detailed knowledge.

An elementaof a semigroupSis calledregularif there existsx∈Ssuch thataxa=a. The semigroupSis calledregularif 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 callSis aninverse semigroupif every element has exactly one inverse. The well-known examples of inverse semigroups are groups and semilattices. An elementa∈S is calledcoregularandbitscoinverseifa=aba=bab. A semigroupSis said to becoregular if each element ofSis coregular []. In addition, letE(S) andRegSbe the set ofidempotent andregularelements, respectively. We then say thatSis calledπ-regularif, for everys∈S, there is anm∈Nsuch thatsm_∈_Reg_S_{. Moreover, if}_S_is_π_{-regular and the set}_E₍_S_{) is a}

commutative subsemigroup ofS, thenSis calledstronglyπ-inverse semigroup[]. We recall thatRegSis an inverse subsemigroup of a stronglyπ-inverse semigroupS.

2 Bruck-Reilly extensions of monoids

Let us suppose thatAis a monoid with an endomorphismθ deﬁned on it such thatAθ is in theH-class [] of the identity AofA. Also, letNdenotes the set of nonnegative

integers. Hence, the setN_×_A_×_N_{with the multiplication}

(m,a,n)m,a,n=m–n+t,aθt–naθt–m,n–m+t,

where t=max(n,m) and θ _{is the identity map on}_A_{, forms a monoid with identity} (, A, ). Then this monoid is called theBruck-Reilly extensionofAdetermined byθ[–]

and denoted byBR(A,θ).

In the above references, the authors usedBR(A,θ) to prove that every semigroup embeds in a simple monoid, and to characterize special classes of inverse semigroups. In [, The-orem .], Munn showed thatBR(A,θ) is an inverse semigroup if and only ifAis inverse. So, the following result is a direct consequence of this theorem.

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Form,m∈N_and_a_,_a_∈_A_{, since}

(m,a,m)m,a,m=m–m+t,aθt–maθt–m,m–m+t= (t,b,t)

with t=max(m,m) andb= (aθt–m₎₍_a_θt–m₎_∈_A_{, the set} _{₍_m_,_a_,_m₎_|_a_∈_A_,_m_∈_N_} be-comes a subsemigroup ofBR(A,θ). Thus, we further have the following lemma.

Lemma  Let(m,a,n)∈BR(A,θ).If(m,a,n)is coregular then m=n.

Proof Let (m,a,n)∈BR(A,θ). Then there exists (m,a,n)∈BR(A,θ) such that

(m,a,n)m,a,n(m,a,n) = (m,a,n) and m,a,n(m,a,n)m,a,n= (m,a,n).

We have

(m,a,n)m,a,n(m,a,n) =m–n–n+m+s,b,n–m+s

for someb∈A, wheres=max(n,m) ands=max(n–m+s,m). This impliesm=m–n– n+m+sandn=n–m+s, in other wordsm+m=n+n. Further, for somec∈A, we have

m,a,n(m,a,n)m,a,n=m–n–n+m+S,c,n–m+S,

where S=max(n,m) andS=max(n–m+S,m). This givesm=m–n–n+m+S, n=n–m+S, and consequently,n=m. Together withm+m=n+n, we obtainm=n

as required.

Lemma  shows that a coregular element inBR(A,θ) and its coinverse belongs to

(m,a,m)|a∈A,m∈N.

Now we can present the following result.

Theorem  Let A be a monoid.Then A={(m,a,m)|a∈A,m∈N_{} ≤}_BR₍_A_,_θ₎_{is coregular} if and only if A is coregular.

Proof Assume thatA≤BR(A,θ) is a coregular monoid. For (,a, )∈BR(A,θ), there ex-ists (m,a,m)∈BR(A,θ) such that

(,a, )m,a,m(,a, ) =m,aθmaaθm,m= (,a, ) ()

and

m,a,m(,a, )m,a,m=m,aaθma,m= (,a, ). ()

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Conversely, let (m,a,m)∈BR(A,θ). Then there is ana∈Awithaaa=aandaaa=a. Thus, for (m,a,m)∈BR(A,θ), we get

(m,a,m)m,a,m(m,a,m) =m,aaa,m= (m,a,m)

and

m,a,m(m,a,m)m,a,m=m,aaa,m= (m,a,m).

Therefore,A={(m,a,m)|a∈A,m∈N_{} ≤}_BR₍_A_,_θ_{) is coregular.}

In [, Theorem .], it is proved that:

• (m,a,n)is an idempotent element inBR(A,θ)if and only ifm=nandais an idempotent element inA.

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

Theorem  BR(A,θ)is stronglyπ-inverse if and only if A is regular and the idempotents in A commute.

Proof LetBR(A,θ) be stronglyπ-inverse, and leta∈A. Also let us consider the element (,a, ) inBR(A,θ). Then there exists an elementr∈Nwith (,a, )r_∈_Reg_BR₍_A_,_θ_{). It is}

actually a routine matter to show that (,a, )r= (,a(aθ)r–,r). Moreover, there exists an elementa∈Asuch that (r,a, ) is an inverse of (,a(aθ)r–_,_r_{) (see []). Therefore,}

,a(aθ)r–,r=,a(aθ)r–,rr,a, ,a(aθ)r–,r

=,a(aθ)r–a,r,a(aθ)r–,r=,a(aθ)r–aa(aθ)r–,r.

This shows that

a(aθ)r–=a(aθ)r–aa(aθ)r–. ()

By the assumption given in the beginning of this section, sinceaθis in theH-class of the element A, we obtainaθis a group element, and so there is an inverse element ((aθ)r–)–.

Thus, by (), we geta=a(aθ)r–_a_a_{; in other words,}_a_∈_Reg_A_{. Consequently,}_A_{is regular.} Now, let us also show that the elements inE(A) are commutative. But this is quite clear by the fact that the idempotents inBR(A,θ) commute if and only if the idempotents inA commute (see [, Theorem .()]).

Conversely, let us suppose thatAis regular and the idempotents inAcommute. Then BR(A,θ) is regular, whereπ-regular by Corollary . Moreover, again by [, Theorem .()], E(BR(A,θ)) is a commutative subsemigroup, which is required toBR(A,θ) satisfy strongly

π-inverse property, hence the result.

3 The generalized Bruck-Reilly∗-extension of monoids

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fromAintoH_∗and, for an elementuinH, letλube the inner automorphism ofH∗deﬁned byx→uxu–such thatγ λu=βγ.

Now one can consider the set S=N_×_N_×_A_×_N_×_N _{into a semigroup with a} multiplication

(m,n,v,p,q)m,n,v,p,q

= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

(m,n–p+d, (vβd–p₎₍_v_βd–n_),_p_–_n₊_d_,_q_), _if_q₌_m_,

(m,n,v(((u–n(vγ)up)γq–m–)βp),p,q–m+q), ifq>m,

(m–q+m,n, (((u–n₍_vγ₎_up₎_γm–q–₎_βn₎_v_,_p_,_q_), _if_q_<_m_,

()

whered=max(p,n) andβ,γare interpreted as the identity map ofA, and alsou_is interpreted as the identity AofA. In [], Yu Shung and Li-Min Wang showed thatSis

a monoid with the identity (, , A, , ). In fact, this new monoidS=N×N×A×

N_×_N_{is denoted by}_GBR∗₍_A_;_β_,_γ_;_u_{) and called}_{generalized Bruck-Reilly}_∗_-_extension_of Adetermined by the morphismsβ,γ and the elementu.

The following lemmas were established in [].

Lemma  If(m,n,v,p,q)∈GBR∗(A;β,γ;u),then(m,n,v,p,q)is an idempotent if and only if m=q,n=p and v is idempotent.

Lemma  If(m,n,v,p,q)∈GBR∗(A;β,γ;u),then(m,n,v,p,q)has an inverse

m,n,v,p,q∈S

if and only if vis an inverse of v in A while m=q,n=p,p=n and q=m.

Then we have an immediate consequence as in the following.

Corollary  Let A be a monoid.Then GBR∗(A;β,γ;u)is regular if and only if A is 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 monoidA, we determine the maximal submonoid ofGBR∗(A;β,γ;u), which can be held coregularity ifAsatisﬁes particular properties.

Our ﬁrst observation is the following.

Lemma  The setL:={(m,n,v,n,m)|v∈A,m,n∈N_}_{is a submonoid of GBR}∗₍_A_;_β_,_γ_;_u_).

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

It turns out that all coregular elements inGBR∗(A;β,γ;u) belong to the submonoidL.

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Proof Let (m,n,v,p,q)∈GBR∗(A;β,γ;u) be a coregular element. Then there exists an el-ement (m,n,v,p,q)∈GBR∗(A;β,γ;u) such that

(m,n,v,p,q)m,n,v,p,q(m,n,v,p,q) = (m,n,v,p,q) and

m,n,v,p,q(m,n,v,p,q)m,n,v,p,q= (m,n,v,p,q).

Letq=mandq=m. Then we have

(m,n,v,p,q)m,n,v,p,q(m,n,v,p,q)

=m,n+n–p–p+z,w,p–n+z,q

for somew∈A, wherez=max(p,n) andz=max(p–n+z,n). This implies that

n=n+n–p–p+z ()

and

p=p–n+z. ()

By (), we haven=z. Applying this in (), we getn+n–p–p+n=nand thusn+n=p+p. Further, we have

m,n,v,p,q(m,n,v,p,q)m,n,v,p,q

=m,n+n–p–p+Z,w,p–n+Z,q

for somew∈A, whereZ=max(p,n) andZ=max(p–n+Z,n). This implies thatm=m, q=q,

n=n+n–p–p+Z ()

and

p=p–n+Z. ()

By writing the equality () in (), we getn=p. Together withn+n=p+p, we obtain n=p. By assumingq=m, we also getm=q.

Now let (x,y,t,z,w)(x,y,t,z,w) = (x,y,t,z,w). Then it is easy to verify thatx≥ x,xandy≥y,y. Ifw=x, we can easily see thatx>x,xory>y,y. This shows that (m,n,a,p,q)(m,n,a,p,q)(m,n,a,p,q)= (m,n,a,p,q) ifq=morq=m. Hence,q=m

orq=mis not possible.

Then we have the following result.

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Proof Suppose thatL={(m,n,v,n,m)|v∈A,m,n∈N_{} ≤}_GBR∗₍_A_;_β_,_γ_;_u_{) is coregular.} For each (m, ,v, ,m) inL, there exists an element (m,n,v,n,m)∈Lsuch that

m, ,v, ,mm,n,v,n,mm, ,v, ,m=m,n,vβnvvβn,n,m

=m, ,v, ,m ()

and

m,n,v,n,mm, ,v, ,mm,n,v,n,m=m,n,vvβnv,n,m

=m, ,v, ,m. ()

By () and (), we obtainn= , and hencevvv=vandvvv=v. So,Ais coregular. Conversely, letAbe a coregular monoid and (m,n,v,n,m)∈L. Then there exists an elementv∈Awithvvv=vandvvv=v. Therefore, for (m,n,v,n,m)∈L, we get

(m,n,v,n,m)m,n,v,n,m(m,n,v,n,m) =m,n,vvv,n,m= (m,n,v,n,m),

m,n,v,n,m(m,n,v,n,m)m,n,v,n,m=m,n,vvv,n,m= (m,n,v,n,m).

Hence,L≤GBR∗(A;β,γ;u) is a coregular monoid, as desired.

In the ﬁnal theorem, we consider stronglyπ-inverse property.

Theorem  GBR∗(A;β,γ;u)is stronglyπ-inverse if and only if A is regular and the idem-potents in A commute.

Proof We will follow the same format as in the proof of Theorem . So, let us suppose thatGBR∗(A;β,γ;u) is a stronglyπ-inverse monoid, and leta∈A. Then, for (, ,a, , )∈ GBR∗(A;β,γ;u), there is an elementr∈Nwith (, ,a, , )r_∈_Reg_GBR∗₍_A_;_β_,_γ_;_u_{). It is}

easily seen that (, ,a, , )r_{= (, ,}_a₍_aβ₎r–_,_r_{, ). Moreover, there is an element}_a_∈_A such that (,r,a, , ) is an inverse of (, ,a(aβ)r–_,_r_{, ) by Lemma . From here, we have}

, ,a(aβ)r–,r, =, ,a(aβ)r–,r, ,r,a, , , ,a(aβ)r–,r, 

=, ,a(aβ)r–aa(aβ)r–,r, .

This actually shows that

a(aβ)r–=a(aβ)r–aa(aβ)r–. ()

At the same time, sinceaβ is in theH∗-class of the A, there exists an inverse element

((aβ)r–₎–_{. Thus, by (), we get}_a₌_a₍_aβ₎r–_a_a_{, in other words,}_a_∈_Reg_A_{. Hence,}_A_is regular. Now, let us show that the elements inE(A) are commutative to conclude the ne-cessity part of the proof. To do that, consider any two elementsvandvinE(A). Thus, (, ,v, , ), (, ,v, , )∈E(GBR∗(A;β,γ;u)) (by Lemma ) and we have

(, ,vv, , ) = (, ,v, , )(, ,v, , )

= (, ,v, , )(, ,v, , ) = (, ,vv, , ).

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Conversely, let us suppose thatAis regular. ThenGBR∗(A;β,γ;u) is regular, whereπ -regular by Corollary . Now we need to show that the elements inE(GBR∗(A;β,γ;u)) com-mute. To do that, let us take (m,n,e,n,m), (m,n,e,n,m)∈E(GBR∗(A;β,γ;u)), and thus ee=eeby Lemma . Now, by considering the multiplication (m,n,e,n,m)(m,n,e,n,m) as deﬁned in (), we have the following cases.

Case (i): Ifm=m, then we get

(m,n,e,n,m)m,n,e,n,m=m,d,eβd–neβd–n,d,m

and

m,n,e,n,m(m,n,e,n,m) =m,d,eβd–neβd–n,d,m,

respectively, where d=max(n,n). Sincee,e∈E(A), we deduce that botheβd–n _and

eβd–n_{are the elements of}_E₍_A_{), in other words,}

eβd–neβd–n=eβd–neβd–n.

Thus, (m,n,e,n,m)(m,n,e,n,m) = (m,n,e,n,m)(m,n,e,n,m). Case (ii): Ifm<morm>m, then we get

(m,n,e,n,m)m,n,e,n,m=m,n,u–n(eγ)unγm–m–βne,n,m,

m,n,e,n,m(m,n,e,n,m) =m,n,eu–n(eγ)unγm–m–βn,n,m

or

(m,n,e,n,m)m,n,e,n,m=m,n,u–neγunγm–m–βne,n,m,

m,n,e,n,m(m,n,e,n,m) =m,n,eu–neγunγm–m–βn,n,m,

respectively. Since ((u–n₍_eγ₎_un₎_γm–m–₎_βn_{, ((}_u–n₍_e_γ₎_un₎_γm–m–₎_βn_∈_E₍_A_{), we clearly}

obtain (m,n,e,n,m)(m,n,e,n,m) = (m,n,e,n,m)(m,n,e,n,m).

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.

Author details

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.

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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doi:10.1186/1687-1812-2013-78