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arXiv · 2006.08499

On separability finiteness conditions in semigroups

Craig Miller·Gerard O'Reilly·Martyn Quick·Nik Ruskuc
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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\"utzenberger 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 $\mathcal{H}$-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 $\mathcal{H}$-classes, we investigate whether it has one of these properties if and only if all its Sch\"utzenberger groups have the property.

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Craig Miller, Gerard O'Reilly, Martyn Quick, Nik Ruskuc. 2020-06-15. On separability finiteness conditions in semigroups. https://arxiv.org/abs/2006.08499

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