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

TL;DR

This paper investigates the word problem in special inverse monoids by linking it to their maximal group images and prefix monoids, providing new decidability conditions and extending previous results to broader classes.

Contribution

It introduces sufficient conditions for the decidability of prefix membership problems in finitely presented groups, enabling algorithmic equivalence of word problems in special inverse monoids and their groups.

Findings

Sufficient conditions for prefix membership problem decidability.

Algorithmic equivalence of word problems in special inverse monoids and their groups.

Extension of results from one-relator groups to arbitrary finitely presented groups.

Abstract

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.