On separability finiteness conditions in semigroups

Craig Miller, Gerard O'Reilly, Martyn Quick, Nik Ruskuc

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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.
Original languageEnglish
Pages (from-to)402-430
JournalJournal of the Australian Mathematical Society
Issue number3
Early online date9 Sept 2021
Publication statusPublished - Dec 2022


  • Separability
  • Finiteness condition
  • Semigroup
  • Congruence
  • Residual finiteness
  • Commutative semigroup
  • Schützenberger group


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