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ON SEPARABILITY FINITENESS CONDITIONS IN SEMIGROUPS
Journal of the Australian Mathematical Society  (IF0.8),  Pub Date : 2021-09-09, DOI: 10.1017/s1446788721000124
CRAIG MILLER, GERARD O’REILLY, MARTYN QUICK, NIK RUŠKUC

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 $\mathcal {H}$ -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 Schützenberger groups have the property.