Jonathan Warne | Searcharxiv

arXiv2025

On the submonoid membership problem for HNN extensions of free groups

We study membership problems in HNN extensions of free groups and then apply these results to solve the word problem in certain families of one-relator inverse monoids. In more detail, we consider HNN extensions where the defining isomorphism produces a bijection between subsets of a basis of the free group. Within such HNN extensions we identify natural conditions on submonoids of this group that suffice for membership in that submonoid to be decidable. We show that these results can then be applied to solve the prefix membership problem in certain one-relator groups which via results of Ivanov, Margolis and Meakin $(2001)$ then give solutions to the word problem for the corresponding one-relator inverse monoid. In particular our new techniques allow us to solve the word problem in an example (Example $7.6$) from Dolinka and Gray $(2021)$ which previous methods had not been able to resolve.

math.GR↗

arXiv2024

The word problem of finitely presented special inverse monoids via their groups of units

A special inverse monoid is one defined by a presentation where all the defining relations have the form $r = 1$. By a result of Ivanov Margolis and Meakin the word problem for such an inverse monoid can often be reduced to the word problem in its maximal group image together with membership in a particular submonoid of that group, called the prefix monoid, being decidable. We prove several results that give sufficient conditions for the prefix membership problem of a finitely presented group to be decidable. These conditions are given in terms of the existence of particular factorisations of the relator words. In particular we are able to find sufficient conditions for a special inverse monoid, its maximal group image and its group of units to have word problems that are algorithmically equivalent. These results extend previous results for one-relator groups to arbitrary finitely presented groups. We then apply these results to solve the word problem in various families of E-unitary special inverse monoids. We also find some criteria for when amalgamations of E-unitary inverse monoids are themselves E-unitary.

math.GR↗