Two applications of monoid actions to cross-sections

TL;DR

This paper uses monoid actions to construct examples demonstrating limitations of regular cross-sections in monoids, answering an open question and exploring the hierarchy of cross-section classes.

Contribution

It introduces a new construction linking monoid actions to cross-section properties, providing counterexamples and hierarchy results in formal language theory.

Findings

A direct product of monoids can have a prefix-closed regular cross-section while a factor does not.

Existence of monoids with cross-sections in one class but not in a larger class.

Counterexamples to the closure properties of regular cross-sections in monoids.

Abstract

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 $\mathfrak{D}$.