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 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.
Citation
Miller , C , O'Reilly , G , Quick , M & Ruskuc , N 2022 , ' On separability finiteness conditions in semigroups ' , Journal of the Australian Mathematical Society , vol. 113 , no. 3 , pp. 402-430 . https://doi.org/10.1017/S1446788721000124
Publication
Journal of the Australian Mathematical Society
Status
Peer reviewed
ISSN
1446-7887Type
Journal article
Rights
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
Description
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.Collections
Items in the St Andrews Research Repository are protected by copyright, with all rights reserved, unless otherwise indicated.