Congruences on Bruck-Reilly extensions of monoids
type: Item
instance of: scholarly article
title: Congruences on Bruck-Reilly extensions of monoids
zbMATH Open document ID: 0828.20050
DOI: 10.1007/BF02573515
author: Brunetto Piochi
published in: Semigroup Forum
publication date: 1995-12-18T00:00:00Z
full work available at URL: https://eudml.org/doc/135377
review text: In 1991, \textit{S. Rankin} started the study of congruences on Bruck-Reilly extensions of inverse monoids [Semigroup Forum 42, No. 3, 279-292 (1991; Zbl 0818.20076)]. In the paper under review the general case is investigated, that is, congruences on Bruck-Reilly extensions \(S=BR(T,\alpha)\) of arbitrary monoids \(T\). It turns out that there are only 3 types of congruences: (i) group-congruences, (ii) congruences \(\rho\) of the form: \((m,a,n)\rho(p,b,q)\) iff \(m=p\), \(n=q\), \((0,a,0)\rho(0,b,0)\), (iii) so-called cross-over congruences. First, all group-congruences on \(S=BR(T,\alpha)\) are described by means of those on \(S_0=BR(G,\alpha)\), where \(G\) denotes the group of units of \(T\). Since \(S_0\) is an inverse semigroup, all group-congruences are known by their kernels. Two alternative descriptions of congruences of type (i) on \(S=BR(T,\alpha)\) are given. Also, the least group-congruence on \(S\) is found to have the same form as that on inverse semigroups. Second, a construction of all cross-over congruences on \(S\) by means of ideals and congruences on \(T\) is given. In particular, it follows that no such congruence exists on \(S\) if \(T\) is simple. Finally, descriptions of the quotient semigroups of \(S\) by a congruence of type (ii) or (iii) as particular Bruck-Reilly semigroups are provided. Note: A summary of these results was published [in \textit{C. Bonzini et al.}, Conference on Semigroups, Luino/Italy 1992, 212-223 (1993; Zbl 0815.20055)].
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 20M15
zbMATH DE Number: 736952
zbMATH Keywords: inverse monoids
zbMATH Keywords: congruences on Bruck-Reilly extensions
zbMATH Keywords: monoids
zbMATH Keywords: group- congruences
zbMATH Keywords: cross-over congruences
zbMATH Keywords: group of units
zbMATH Keywords: kernels
zbMATH Keywords: inverse semigroups
zbMATH Keywords: quotient semigroups
zbMATH Keywords: Bruck-Reilly semigroups
reviewed by: H. Mitsch
MaRDI profile type: Publication
cites work: On a sublattice of the lattice of congruences on a simple regular ω-semigroup
cites work: Congruences on simple regular ω-semigroups
cites work: The Least Commutative Congruence on a simple regular ω-semigroup
cites work: Q5515384
cites work: On Bruck-Reilly extensions
cites work: Q3032500
cites work: Regular ω-semigroups
cites work: Congruences on a bisimple ω-semigroup
cites work: Congruences on simple ω-semigroups
cites work: Q3337662
cites work: Congruences on an inverse Bruck-Reilly monoid. I: Non-group congruences
cites work: Bisimple ω-Semigroups
Recommended article: Congruences on an inverse Bruck-Reilly monoid. I: Non-group congruences
Recommended article: Q4311871
Recommended article: Presentations of Bruck-Reilly extensions and decision problems
Recommended article: Q4527147
Recommended article: On Bruck-Reilly extensions of rectangular bands, zero semigroups and free monoids.
Recommended article: The Congruence Lattice of Bruck–Reilly Extensions
Recommended article: Finite presentability of Bruck-Reilly extensions of groups
Recommended article: On Finite Presentability of Bruck–Reilly Extensions of a Monoid with Respect to an Endomorphism and Its Powers
Recommended article: Some fixed-point results on (generalized) Bruck-Reilly \(*\)-extensions of monoids.
Recommended article: A relation for general and inverse semigroups.
description: scientific article; zbMATH DE number 736952