Residual finiteness and related properties in monounary algebras and their direct products
Bill de Witt · Algebra Universalis · 2021
Abstract 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.