  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 $\mathfrak{D}$.
