Minimal presentations and efficiency of semigroups
type: Item
instance of: scholarly article
title: Minimal presentations and efficiency of semigroups
zbMATH Open document ID: 0958.20040
author: Hayrullah Ayık
author: J. J. O'Connor
author: N. Ruškuc
author: C. M. Campbell
published in: Semigroup Forum
publication date: 2001-04-17T00:00:00Z
review text: Let \(A\) be an alphabet and let \(A^+\) be the free semigroup generated by \(A\). A semigroup presentation is an ordered pair \({\mathcal P}=\langle A|R\rangle\), where \(R\subset A^+\times A^+\). The deficiency \(\text{def}({\mathcal P})\) of \(\mathcal P\) is the number \(|R|-|A|\). The deficiency \(\text{def}(S)\) of a finitely presented semigroup \(S\) is defined to be \(\min\{\text{def}({\mathcal P})\mid{\mathcal P}\) is a finite presentation of \(S\}\). In general, \(\text{def}(S)\geq\text{rank}(H_2(S))\) for any finite semigroup \(S\), where \(H_2(S)=\text{Tor}^{\mathbb{Z} S}_2(\mathbb{Z},\mathbb{Z})\) is the second homology group of \(S\). A finite semigroup \(S\) is efficient if \(\text{def}(S)=\text{rank}(H_2(S))\). The authors show that finite Abelian groups, the dihedral groups \(D_{2n}\) with \(n\) even, and finite rectangular bands are efficient. On the other hand, finite zero semigroups and finite semilattices, though they are Abelian, are not efficient.
reviewed by: Yuji Kobayashi
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 1423170
zbMATH Keywords: efficiency
zbMATH Keywords: semigroup presentations
zbMATH Keywords: deficiency
zbMATH Keywords: finitely presented semigroups
zbMATH Keywords: finite semigroups
zbMATH Keywords: homology groups
zbMATH Keywords: finite Abelian groups
zbMATH Keywords: dihedral groups
zbMATH Keywords: finite rectangular bands
zbMATH Keywords: finite zero semigroups
zbMATH Keywords: finite semilattices
MaRDI profile type: Publication
full work available at URL: s002339910016
OpenAlex ID: W1980853405
DOI: 10.1007/S002339910016
description: scientific article; zbMATH DE number 1423170