Presentations for subgroups of monoids
type: Item
instance of: scholarly article
title: Presentations for subgroups of monoids
zbMATH Open document ID: 0943.20057
author: N. Ruškuc
published in: Journal of Algebra
publication date: 2000-09-04T00:00:00Z
review text: The author generalizes the Reidemeister-Schreier result for subgroup presentations to presentations of subgroups of monoids. For a subset \(X\subseteq S\) of a monoid \(S\) and an element \(s\in S\), \(Xs\) is a coset if there exists an element \(t\in S\) such that \(Xst=X\). The index \([S:X]\) of \(X\) (in \(S\)) is the number of cosets of \(X\) in \(S\). Now \(S\) acts on the set \(I\cup\{0\}\) by \(C_is=C_{is}\), where \(C_i\) (\(i\in I\), \(0\not\in I\)) are the cosets of \(X\), and \(C_0\) satisfies \(C_is=C_0\) if \(C_is\) is not a coset, and \(C_0s=C_0\). Let \(G\) be a subgroup (with identity \(e\)) of a monoid \(S\). Then a set of elements \(r_i,r_i'\in S\) with \(i\in I\) is a system of coset representatives if \(Gr_i=C_i\), \(gr_ir_i'=g\) for all \(i\in I\) and \(g\in G\) (and \(r_1=r_1'=1\)). It is proved that if \(S\) is generated by \(A\), then \(Y=\{er_iar_{ia}'\mid i\in I,\;a\in A,\;ia\neq 0\}\) generates \(G\) as a monoid. Using a system of coset representatives, a monoid presentation is constructed for a subgroup \(G\) from a presentation of the monoid \(S\). As a corollary we have that a subgroup of finite index in a finitely presented monoid is finitely presented. The author also considers applications to the group of units of a monoid, and to regular monoids with finitely many left and right ideals.
reviewed by: Tero Harju
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 1356306
zbMATH Keywords: subgroup presentations
zbMATH Keywords: subgroups of monoids
zbMATH Keywords: numbers of cosets
zbMATH Keywords: coset representatives
zbMATH Keywords: monoid presentations
zbMATH Keywords: subgroups of finite index
zbMATH Keywords: finitely presented monoids
zbMATH Keywords: groups of units
zbMATH Keywords: regular monoids
MaRDI profile type: Publication
full work available at URL: jabr.1999.7930
OpenAlex ID: W2027874150
cites work: A Remark on Generalised Free Products
cites work: Reidemeister-Schreier type rewriting for semigroups
cites work: REWRITING A SEMIGROUP PRESENTATION
cites work: Semigroup and Group Presentations
cites work: On subsemigroups of finitely presented semigroups
cites work: Presentations for subsemigroups -- applications to ideals of semigroups
cites work: On the Left and Right Cohomological Dimension of Monoids
cites work: Q4846425
cites work: Constructions and presentations for monoids
cites work: Complete rewriting systems and homology of monoid algebras
cites work: Q3853827
cites work: Q4145882
cites work: Q5513066
cites work: Q5465055
cites work: LOW-DIMENSIONAL HOMOTOPY THEORY FOR MONOIDS
cites work: Bisimple ω-Semigroups
cites work: Structure Results for Transitive, Untwisted, Superlinked Finite Covers
cites work: Syntactic and Rees indices of subsemigroups
cites work: Word problems and a homological finiteness condition for monoids
cites work: Free inverse semigroups are not finitely presentable
cites work: Clifford monoid presentations
cites work: Presentations of inverse monoids
cites work: Finite complete rewriting systems and finite derivation type for small extensions of monoids
cites work: Applying rewriting methods to special monoids
DOI: 10.1006/JABR.1999.7930
description: scientific article; zbMATH DE number 1356306