Groups and actions in transformation semigroups
type: Item
instance of: scholarly article
title: Groups and actions in transformation semigroups
zbMATH Open document ID: 0902.20028
DOI: 10.1007/PL00004628
author: K. Appert
published in: Mathematische Zeitschrift
publication date: 1998-12-06T00:00:00Z
review text: Let \(S\) be a transformation semigroup of degree \(n\). To each element \(s\in S\) a permutation group \(G_R(s)\), acting on the image of \(s\), is associated. A natural generating set for this group is found. It turns out that the \(\mathcal R\)-class of \(s\) is a disjoint union of certain sets, each having size equal to the size of \(G_R(s)\). As a consequence, it is shown that two \(\mathcal R\)-classes containing elements with equal images have the same size, even if they do not belong to the same \(\mathcal D\)-class. By a certain duality process one associates to \(s\) another permutation group \(G_L(s)\) on the image of \(s\), and proves analogous results for the \(\mathcal L\)-class of \(S\). Finally, it is proved that the Schützenberger group of the \(\mathcal H\)-class of \(s\) is isomorphic to the intersection of \(G_R(s)\) and \(G_L(s)\). The results of this paper can also be applied in new algorithms for investigating transformation semigroups, which will be described in a forthcoming paper.
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20B40
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 1197139
zbMATH Keywords: actions
zbMATH Keywords: orbits
zbMATH Keywords: Green's relations
zbMATH Keywords: permutation groups
zbMATH Keywords: generating sets
zbMATH Keywords: duality
zbMATH Keywords: Schützenberger groups
zbMATH Keywords: algorithms
zbMATH Keywords: transformation semigroups
MaRDI profile type: Publication
full work available at URL: pl00004628
OpenAlex ID: W1979501573
description: scientific article; zbMATH DE number 1197139