Two applications of monoid actions to cross-sections

Abstract

Using a construction that builds a monoid from a monoid action, this article exhibits an example of a direct product of monoids that admits a prefix-closed regular cross-section, but one of whose factors does not admit a regular cross-section; this answers negatively an open question from the theory of Markov monoids. The same construction is then used to show that for any full trios and such that is not a subclass of there is a monoid with a cross-section in but no cross-section in