Iteration of expansions. Unambiguous semigroups
type: Item
instance of: scholarly article
title: Iteration of expansions. Unambiguous semigroups
zbMATH Open document ID: 0547.20054
DOI: 10.1016/0022-4049(84)90055-0
author: J.-C. Birget
published in: Journal of Pure and Applied Algebra
publication date: 1984-01-01T00:00:00Z
review text: In this paper the author introduces a semigroup expansion which simultaneously enjoys properties of the left and right Rhodes expansions which is used in another paper [reviewed below]. He begins with \(\hat S_ A^{{\mathcal L}}\), the left Rhodes expansion cut down to generators which is defined as \(<(a)| a\in A>_{\hat S^{{\mathcal L}}}\) where \(S=\) and \(\hat S^{{\mathcal L}}\) is the ordinary left Rhodes expansion. \(\hat S^{{\mathcal L}}_ A\) has unambiguous \({\mathcal L}\)-order (i.e. if y and z are \({\mathcal L}\)-above x, then y and z are \({\mathcal L}\)-comparable) and the \(h_{{\mathcal L}}\)-property (i.e. for each \({\mathcal L}\)-class \(L_ 1\) of \(\hat S^{{\mathcal L}}_ A\) there is a finite bound on the length of ascending \({\mathcal L}\)-chains from \(L_ 1)\). \(\hat S^{{\mathcal R}}_ A\) is defined dually. The main idea is to iterate these expansions alternately (iteration of the same expansion has no effect). This leads to two infinite chains of expansions and canonical morphisms the projective limits of which are isomorphic. This expansion is denoted by \(\hat S^+_ A\) and significantly the defining projective limit is finitary, meaning that given \((s_ n)_{n\in\omega }\in\hat S^+_ A\) it is possible to determine the equality of \((s_ n)_{n\in\omega }\) to any other member of \(\hat S^+_ A\) by comparison of the first N members of each sequence (N depending only on \((s_ n)_{n\in\omega })\). This leads to the main theorem which in part says: for any semigroup S, generated by a subset A, there exists a semigroup expansion \(\hat S^+_ A\), generated by a subset of the same cardinality \(| A|\), also denoted by A, and a surmorphism \(\eta:\widehat S^+_ A\twoheadrightarrow S\), injective on A such that (1) \(\hat S^+_ A\) is invariant under the expansions (\({\hat\cdot })^{{\mathcal L}}_ A\) and (\({\hat\cdot })^{{\mathcal R}}_ A\). Hence \(\hat S^+_ A\) has unambiguous \({\mathcal L}\)- and \({\mathcal R}\)- order and has the \(h_{{\mathcal L}}…
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M15
zbMATH DE Number: 3871665
zbMATH Keywords: right Rhodes expansions
zbMATH Keywords: generators
zbMATH Keywords: length of ascending \({\mathcal L}\)-chains
zbMATH Keywords: infinite chains of expansions
zbMATH Keywords: projective limits
zbMATH Keywords: semigroup expansion
zbMATH Keywords: regular \({\mathcal D}\)-class
zbMATH Keywords: regular elements
zbMATH Keywords: \({\mathcal J}\)-class
reviewed by: Peter M. Higgins
MaRDI profile type: Publication
full work available at URL: 0022-4049(84)90055-0
OpenAlex ID: W2077630085
cites work: Q5570933
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cites work: Infinite iteration of matrix semigroups. II: Structure theorem for arbitrary semigroups up to aperiodic morphism
description: scientific article; zbMATH DE number 3871665