ON SEPARABILITY FINITENESS CONDITIONS IN SEMIGROUPS

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

We investigate four finiteness conditions related to residual finiteness: complete separability, strong subsemigroup separability, weak subsemigroup separability and monogenic subsemigroup separability. For each of these properties we examine under which conditions the property is preserved under direct products. We also consider if any of the properties are inherited by the factors in a direct product. We give necessary and sufficient conditions for finite semigroups to preserve the properties of strong subsemigroup separability and monogenic subsemigroup separability in a direct product.

Residual finiteness and related properties in monounary algebras and their direct products

In this paper we discuss the relationship between direct products of monounary algebras and their components, with respect to the properties of residual finiteness, strong/weak subalgebra separability, and complete separability. For each of these properties $${\mathcal {P}}$$ P , we give a criterion $$\mathcal {C_P}$$ C P such that a monounary algebra $$A$$ A has property $${\mathcal {P}}$$ P if and only if it satisfies $$\mathcal {C_P}$$ C P . We also show that for a direct product $$A\times B$$ A × B of monounary algebras, $$A\times B$$ A × B has property $${\mathcal {P}}$$ P if and only if one of the following is true: either both $$A$$ A and $$B$$ B have property $${\mathcal {P}}$$ P , or at least one of $$A$$ A or $$B$$ B are backwards-bounded, a special property which dominates direct products and which guarantees all $${\mathcal {P}}$$ P hold.

Residually finite tubular groups

A tubular group G is a finite graph of groups with ℤ2 vertex groups and ℤ edge groups. We characterize residually finite tubular groups: G is residually finite if and only if its edge groups are separable. Methods are provided to determine if G is residually finite. When G has a single vertex group an algorithm is given to determine residual finiteness.