ON SEPARABILITY FINITENESS CONDITIONS IN SEMIGROUPS
Craig J. Miller, Gerard O'Reilly, Martyn Quick, Nik Ruškuc · Journal of the Australian Mathematical Society · 2021
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 semigroupS, these three separability conditions coincide and are equivalent to every $\mathcal {H}$ -class ofSbeing 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.