Green index in semigroups: generators, presentations, and automatic structures.
type: Item
instance of: scholarly article
title: Green index in semigroups: generators, presentations, and automatic structures.
zbMATH Open document ID: 1270.20059
author: Alan J. Cain
author: N. Ruškuc
author: R. Gray
published in: Semigroup Forum
publication date: 2013-03-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/0912.1266
full work available at URL: http://hdl.handle.net/10023/2760
review text: Let \(T\) be a subsemigroup of a semigroup \(S\). \(T\)-relative Green relations [\textit{A. D. Wallace}, Acta Math. Acad. Sci. Hung. 14, 137-148 (1963; Zbl 0122.26802)] are defined for \(u,v\in S\) by \(u\mathcal R^Tv\Leftrightarrow uT^1=vT^1\), \(u\mathcal L^Tv\Leftrightarrow T^1u=T^1v\), \(\mathcal H^T=\mathcal R^T\cap\mathcal L^T\). The Green index of \(T\) in \(S\) is (the number of \(T\)-relative \(\mathcal H^T\)-classes in \(S\setminus T)+1\), i.e. the number of strong orbits in \(S\setminus T\) under the actions of \(T\) on \(S\). For each \(T\)-relative \(\mathcal H^T\)-class \(H\) the stabilizer of \(H\) in \(T\) is \(\text{Stab}(H)=\{t\in T^1:Ht=H\}\) and the relative Schützenberger group of \(H\) is \(\text{Stab}(H)/\gamma\), \((x,y)\in\gamma\Leftrightarrow (\forall h\in H)\;(hx=hy)\). Here is proved a rewriting theorem which shows how generating sets for \(S\) may be used to obtain generating sets for \(T\) and the Schützenberger groups and vice versa and presented a method for constructing a presentation for \(S\) from presentations of \(T\) and the Schützenberger groups. This allows to show that passing to finite Green index subsemigroups or extensions preserves finite generation, solubility of the word problem, growth type, automaticity for subsemigroups, finite presentability for extensions and finite Malcev presentability for group-embeddable semigroups.
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20M35
Mathematics Subject Classification ID: 68Q45
zbMATH DE Number: 6141003
zbMATH Keywords: Green index
zbMATH Keywords: presentations
zbMATH Keywords: generating sets
zbMATH Keywords: orbits
zbMATH Keywords: Schützenberger groups
zbMATH Keywords: rewriting theorems
zbMATH Keywords: automatic semigroups
zbMATH Keywords: finiteness conditions
zbMATH Keywords: Green relations
reviewed by: Jaak Henno
MaRDI profile type: Publication
OpenAlex ID: W1998095289
arXiv ID: 0912.1266
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DOI: 10.1007/S00233-012-9406-2
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