Reidemeister-Schreier type rewriting for semigroups
type: Item
instance of: scholarly article
title: Reidemeister-Schreier type rewriting for semigroups
zbMATH Open document ID: 0833.20069
DOI: 10.1007/BF02573619
author: N. Ruškuc
author: C. M. Campbell
author: E. F. Robertson
author: Richard M. Thomas
published in: Semigroup Forum
publication date: 1996-03-17T00:00:00Z
full work available at URL: https://eudml.org/doc/135399
review text: Let \(T\) be a subsemigroup of a semigroup \(S\). The index of \(T\) in \(S\) is defined as \(1+\text{card}(S-T)\). The authors give a rewriting method that yields a presentation for \(T\) from a given presentation for \(S\). This general method is not constructive. However, when \(T\) is a two sided ideal of \(S\), the authors produce a rewriting procedure similar to the Reidemeister-Schreier rewriting theorem for groups. The methods of proof have similarities with those for groups. However, the presentation and the proof are rather different and technically more complicated than those for groups. For any semigroup \(S\), \(\text{rank}(S)\) is the least cardinality of a generating set for \(S\). Corollaries of the authors' presentations are: For any subsemigroup \(T\) of index \(i\) in a semigroup \(S\), \(\text{rank}(T)\leq i^2\text{rank}(S)\). If \(S\) is finitely presented and \(T\) is a two sided ideal of \(S\) of finite index, then \(T\) is finitely presented. The authors study also presentations for the minimal ideal of a semigroup \(S\) and its Schützenberger group. Suppose \(I\) is a 0-completely simple minimal 0-ideal of a finitely presented semigroup \(S\). If \(I\) has finitely many 0-minimal left ideals or finitely many 0-minimal right ideals, then its Schützenberger group is finitely presented. If \(I\) has finitely many 0-minimal left ideals and finitely many 0-minimal right ideals, then \(I\) is finitely presented.
reviewed by: Awad A. Iskander
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20M35
Mathematics Subject Classification ID: 20M12
Mathematics Subject Classification ID: 68Q42
zbMATH DE Number: 784130
zbMATH Keywords: index
zbMATH Keywords: rewriting methods
zbMATH Keywords: Reidemeister-Schreier rewriting method
zbMATH Keywords: generating sets
zbMATH Keywords: two sided ideals
zbMATH Keywords: presentations
zbMATH Keywords: Schützenberger groups
zbMATH Keywords: finitely presented semigroups
zbMATH Keywords: 0-minimal left ideals
MaRDI profile type: Publication
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OpenAlex ID: W2002252248
description: scientific article; zbMATH DE number 784130