Using a construction that builds a monoid from a monoid action, this paper 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 $\mathfrak{ C}$ and $\mathfrak{ D}$ such that $\mathfrak{ C}$ is not a subclass of $\mathfrak{ D}$, there is a monoid with a cross-section in $\mathfrak{ C}$ but no cross-section in $\mathfrak{ D}$.
Categories: math.GR