On separability finiteness conditions in semigroups
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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.
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
Journal of the Australian Mathematical Society
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
DescriptionFunding: 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.
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