On separability finiteness conditions in semigroups

Craig Miller; Gerard O'Reilly; Martyn Quick; Nik Ruskuc

doi:10.1017/S1446788721000124

On separability finiteness conditions in semigroups

Craig Miller, Gerard O'Reilly, Martyn Quick, Nik Ruskuc

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

Abstract

Taking residual finiteness as a starting point, we consider three related finiteness properties: weak subsemigroup separability, strong subsemigroup separability and complete separability. We investigate whether each of these properties is inherited by Schützenberger groups. The main result of this paper states that for a finitely generated commutative semigroup S, these three separability conditions coincide and are equivalent to every H -class of S being finite. We also provide examples to show that these properties in general differ for commutative semigroups and finitely generated semigroups. For a semigroup with finitely many H -classes, we investigate whether it has one of these properties if and only if all its Schützenberger groups have the property.

Original language	English
Pages (from-to)	402-430
Journal	Journal of the Australian Mathematical Society
Volume	113
Issue number	3
Early online date	9 Sept 2021
DOIs	https://doi.org/10.1017/S1446788721000124
Publication status	Published - Dec 2022

Keywords

Separability
Finiteness condition
Semigroup
Congruence
Residual finiteness
Commutative semigroup
Schützenberger group

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10.1017/S1446788721000124Licence: CC BY

Miller-2021-On-separability-finiteness-JAMS-CCBY
Copyright © Australian Mathematical Publishing Association Inc. 2021. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/ licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited
Final published version, 475 KBLicence: CC BY

Cite this

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title = "On separability finiteness conditions in semigroups",

abstract = "Taking residual finiteness as a starting point, we consider three related finiteness properties: weak subsemigroup separability, strong subsemigroup separability and complete separability. We investigate whether each of these properties is inherited by Sch{\"u}tzenberger groups. The main result of this paper states that for a finitely generated commutative semigroup S, these three separability conditions coincide and are equivalent to every H -class of S being finite. We also provide examples to show that these properties in general differ for commutative semigroups and finitely generated semigroups. For a semigroup with finitely many H -classes, we investigate whether it has one of these properties if and only if all its Sch{\"u}tzenberger groups have the property.",

keywords = "Separability, Finiteness condition, Semigroup, Congruence, Residual finiteness, Commutative semigroup, Sch{\"u}tzenberger group",

author = "Craig Miller and Gerard O'Reilly and Martyn Quick and Nik Ruskuc",

note = "Funding: The first author is grateful to EPSRC for financial support. The second author is grateful to the School of Mathematics and Statistics of the University of St Andrews for financial support.",

year = "2022",

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doi = "10.1017/S1446788721000124",

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TY - JOUR

T1 - On separability finiteness conditions in semigroups

AU - Miller, Craig

AU - O'Reilly, Gerard

AU - Quick, Martyn

AU - Ruskuc, Nik

N1 - Funding: The first author is grateful to EPSRC for financial support. The second author is grateful to the School of Mathematics and Statistics of the University of St Andrews for financial support.

PY - 2022/12

Y1 - 2022/12

N2 - Taking residual finiteness as a starting point, we consider three related finiteness properties: weak subsemigroup separability, strong subsemigroup separability and complete separability. We investigate whether each of these properties is inherited by Schützenberger groups. The main result of this paper states that for a finitely generated commutative semigroup S, these three separability conditions coincide and are equivalent to every H -class of S being finite. We also provide examples to show that these properties in general differ for commutative semigroups and finitely generated semigroups. For a semigroup with finitely many H -classes, we investigate whether it has one of these properties if and only if all its Schützenberger groups have the property.

AB - Taking residual finiteness as a starting point, we consider three related finiteness properties: weak subsemigroup separability, strong subsemigroup separability and complete separability. We investigate whether each of these properties is inherited by Schützenberger groups. The main result of this paper states that for a finitely generated commutative semigroup S, these three separability conditions coincide and are equivalent to every H -class of S being finite. We also provide examples to show that these properties in general differ for commutative semigroups and finitely generated semigroups. For a semigroup with finitely many H -classes, we investigate whether it has one of these properties if and only if all its Schützenberger groups have the property.

KW - Separability

KW - Finiteness condition

KW - Semigroup

KW - Congruence

KW - Residual finiteness

KW - Commutative semigroup

KW - Schützenberger group

U2 - 10.1017/S1446788721000124

DO - 10.1017/S1446788721000124

M3 - Article

SN - 1446-7887

VL - 113

SP - 402

EP - 430

JO - Journal of the Australian Mathematical Society

JF - Journal of the Australian Mathematical Society

IS - 3

ER -

On separability finiteness conditions in semigroups

Abstract

Keywords

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