On residual finiteness of monoids, their Schützenberger groups and associated actions

R Gray; Nik Ruskuc

doi:10.1016/j.jalgebra.2014.02.025

On residual finiteness of monoids, their Schützenberger groups and associated actions

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we discuss connections between the following properties: (RFM) residual finiteness of a monoid M ; (RFSG) residual finiteness of Schützenberger groups of M ; and (RFRL) residual finiteness of the natural actions of M on its Green's R- and L-classes. The general question is whether (RFM) implies (RFSG) and/or (RFRL), and vice versa. We consider these questions in all the possible combinations of the following situations: M is an arbitrary monoid; M is an arbitrary regular monoid; every J-class of M has finitely many R- and L-classes; M has finitely many left and right ideals. In each case we obtain complete answers, which are summarised in a table.

Original language	English
Pages (from-to)	21-45
Journal	Journal of Algebra
Volume	407
Early online date	25 Mar 2014
DOIs	https://doi.org/10.1016/j.jalgebra.2014.02.025
Publication status	Published - 1 Jun 2014

Keywords

Residual fitness
Schützenberger group
Monoid

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10.1016/j.jalgebra.2014.02.025

GrayRuskucResidualFinitenessRevised
Copyright 2014, Elsevier. This is the author’s version of a work that was accepted for publication in Journal of Algebra. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Algebra, Vol 407, June 2014. DOI: http://dx.doi.org/10.1016/j.jalgebra.2014.02.025
Accepted author manuscript, 348 KB

Theory of Semigroups: Representation theory of Semigroups
Ruskuc, N. (PI)
EPSRC
1/04/12 → 30/09/15
Project: Standard
Finiteness Conditions and Index: Finiteness Conditions and Index in Semigroups and Monoids
Gray, R. D. (PI)
EPSRC
1/02/08 → 31/01/11
Project: Standard

Nik Ruskuc
- nik.ruskucst-andrews.acuk
- School of Mathematics and Statistics - Director of Research
- Pure Mathematics - Professor
- Centre for Interdisciplinary Research in Computational Algebra
Person: Academic

Cite this

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title = "On residual finiteness of monoids, their Sch{\"u}tzenberger groups and associated actions",

abstract = "In this paper we discuss connections between the following properties: (RFM) residual finiteness of a monoid M ; (RFSG) residual finiteness of Sch{\"u}tzenberger groups of M ; and (RFRL) residual finiteness of the natural actions of M on its Green's R- and L-classes. The general question is whether (RFM) implies (RFSG) and/or (RFRL), and vice versa. We consider these questions in all the possible combinations of the following situations: M is an arbitrary monoid; M is an arbitrary regular monoid; every J-class of M has finitely many R- and L-classes; M has finitely many left and right ideals. In each case we obtain complete answers, which are summarised in a table.",

keywords = "Residual fitness, Sch{\"u}tzenberger group, Monoid",

author = "R Gray and Nik Ruskuc",

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AU - Gray, R

AU - Ruskuc, Nik

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AB - In this paper we discuss connections between the following properties: (RFM) residual finiteness of a monoid M ; (RFSG) residual finiteness of Schützenberger groups of M ; and (RFRL) residual finiteness of the natural actions of M on its Green's R- and L-classes. The general question is whether (RFM) implies (RFSG) and/or (RFRL), and vice versa. We consider these questions in all the possible combinations of the following situations: M is an arbitrary monoid; M is an arbitrary regular monoid; every J-class of M has finitely many R- and L-classes; M has finitely many left and right ideals. In each case we obtain complete answers, which are summarised in a table.

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KW - Schützenberger group

KW - Monoid

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DO - 10.1016/j.jalgebra.2014.02.025

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SN - 0021-8693

VL - 407

SP - 21

EP - 45

JO - Journal of Algebra

JF - Journal of Algebra

ER -

On residual finiteness of monoids, their Schützenberger groups and associated actions

Abstract

Keywords

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Projects

Theory of Semigroups: Representation theory of Semigroups

Finiteness Conditions and Index: Finiteness Conditions and Index in Semigroups and Monoids

Profiles

Nik Ruskuc

Cite this