Two applications of monoid actions to cross-sections

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 (Formula presented.) and (Formula presented.) such that (Formula presented.) is not a subclass of (Formula presented.) there is a monoid with a cross-section in (Formula presented.) but no cross-section in (Formula presented.).