On Separability Finiteness Conditions in Semigroups

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 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.

On separability properties in direct products of semigroups

“…In [2] Allenby and Gregorac attribute this result to Mal'cev in [9]. It was shown in [10,Lemma 2.4] that a group is completely separable if and only if it is finite. Hence, trivially, the direct product of two CS groups is itself CS.…”

“…For groups G and H, both G and H are always isomorphic to subgroups of G × H. As subgroups inherit all four of these properties, an easy extension of [10,Proposition 2.3], if G × H has one of these properties then so will both G and H. However, the situation for algebras in general, and for semigroups in particular, may not always be so straightforward. Indeed, in [13] de Witt was able to show that there exist monounary algebras A and B such that A is not residually finite but A × B is completely separable.…”

“…The first case is that T contains an idempotent. Then S is isomorphic to a subsemigroup of the SSS semigroup S × T and hence S must also be SSS by [10,Proposition 2.3].…”

“…Let G be an infinite SSS group (e.g. [10,Example 5.4]). Let U ≤ G × P be generated by the set {(g, p) | g ∈ G \ {1 G }}.…”

“…Hence we can separate (x, y) from {(s k , t k ) | k ≥ i + 1} by factoring through S and using the weak subsemigroup separability of S. That is, there exists a finite semigroup P 1 and a homomorphism φ 1 : S × T → P 1 such that φ 1 (x, y) = φ 1 (s n , t n ) for all n ≥ i + 1. As S and T are both WSS, they are both residually finite by [10,Proposition 2.1]. Hence S × T is residually finite by [6,Theorem 2].…”