Word Problem Languages for Free Inverse Monoids

TL;DR

This paper investigates the formal language complexity of word problems in free inverse monoids, revealing their classification within various language classes and establishing limitations based on rank.

Contribution

It provides a detailed classification of the word problem complexities for free inverse monoids, including new results on their context-free and ET0L properties.

Findings

No free inverse monoid has a context-free word problem.

The rank 1 free inverse monoid's word problem is 2-context-free and ET0L.

The co-word problem of rank 1 free inverse monoid is context-free.

Abstract

This paper considers the word problem for free inverse monoids of finite rank from a language theory perspective. It is shown that no free inverse monoid has context-free word problem; that the word problem of the free inverse monoid of rank $1$ is both $2$-context-free (an intersection of two context-free languages) and ET0L; that the co-word problem of the free inverse monoid of rank $1$ is context-free; and that the word problem of a free inverse monoid of rank greater than $1$ is not poly-context-free.