Residual Finiteness and Related Properties in Monounary Algebras and their Direct Products

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 P, we give a graphical criterion C P such that a monounary algebra A has property P if and only if it satisfies C P . We also show that for a direct product A × B of monounary algebras, A × B has property P if and only if one of the following is true: either both A and B have property P, or at least one of A or B are backwards-bounded, a special property which dominates direct products and which guarantees all P hold.

On separability properties in direct products of semigroups

“…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. So factors of a direct product of monounary algebras which has separability property P need not be P themselves.…”