Green index and finiteness conditions for semigroups.
type: Item
instance of: scholarly article
title: Green index and finiteness conditions for semigroups.
zbMATH Open document ID: 1172.20045
published in: Journal of Algebra
publication date: 2008-12-05T00:00:00Z
review text: Given a semigroup \(S\) and a subsemigroup \(T\), the Rees index of \(T\) in \(S\) is defined to be the cardinality of the complement \(S\setminus T\). In this article, in Section 2, the authors give basic properties of Green index. Then in Section 3 and Section 4, they prove the following main results of their paper: Let \(S\) be a semigroup and let \(T\) be a subsemigroup of \(S\) with finite Green index. Let \(\Gamma_i\) (\(i\in I\)) be the Schützenberger groups of the \(T\)-relative \(H\)-classes of the complement \(S\setminus T\). Then the following hold: (I) \(S\) is locally finite if and only if \(T\) is locally finite, in which case every group \(\Gamma_i\) is locally finite; (II) \(S\) is periodic if and only if \(T\) is periodic, in which case every group \(\Gamma_i\) is periodic; (III) \(S\) has finitely many right ideals if and only if \(T\) has finitely many right ideals (and the dual result for left ideals); (IV) \(S\) is residually finite if and only if \(T\) and \(\Gamma_i \) are all residually finite. In Section 5, the authors define the relationship between Green index and syntactic index. Finally, they give some applications of their results.
Mathematics Subject Classification ID: 20M10
zbMATH DE Number: 5378359
zbMATH Keywords: Rees index
zbMATH Keywords: Green index
zbMATH Keywords: Schützenberger groups
zbMATH Keywords: locally finite semigroups
zbMATH Keywords: periodic semigroups
zbMATH Keywords: right ideals
zbMATH Keywords: residually finite semigroups
zbMATH Keywords: syntactic index
zbMATH Keywords: finiteness conditions
author: R. Gray
author: N. Ruškuc
reviewed by: Firat Ateş
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.07.008
OpenAlex ID: W2048063555
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DOI: 10.1016/J.JALGEBRA.2008.07.008
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description: scientific article; zbMATH DE number 5378359