On the rational subsets of the monogenic free inverse monoid

TL;DR

This paper investigates rational subsets of the monogenic free inverse monoid, establishing decidability results for the equality and recognition problems, and characterizing rational submonoids as finitely generated.

Contribution

It proves the decidability of the equality and recognition problems for rational subsets and characterizes rational submonoids as finitely generated within the monogenic free inverse monoid.

Findings

Equality problem for rational subsets is decidable.

Recognition problem for rational subsets is decidable.

Rational submonoids are exactly the finitely generated submonoids.

Abstract

We prove that the equality problem is decidable for rational subsets of the monogenic free inverse monoid $F$. It is also decidable whether or not a rational subset of $F$ is recognizable. We prove that a submonoid of $F$ is rational if and only if it is finitely generated. We also prove that the membership problem for rational subsets of a finite $\mathcal{J}$-above monoid is decidable, covering the case of free inverse monoids.