On regularity and the word problem for free idempotent generated semigroups

TL;DR

This paper investigates the word problem for free idempotent generated semigroups, providing conditions for decidability in finite cases and demonstrating undecidability in certain structures, linking it to subgroup membership problems.

Contribution

It introduces new results on the decidability of the word problem for free idempotent generated semigroups, including both positive and negative findings.

Findings

Decidability of regular element recognition for finite biordered sets.

Decidability of the word problem for regular words under certain subgroup conditions.

Existence of a biorder with an undecidable word problem despite decidable maximal subgroups.

Abstract

The category of all idempotent generated semigroups with a prescribed structure $\mathcal{E}$ of their idempotents $E$ (called the biordered set) has an initial object called the free idempotent generated semigroup over $\mathcal{E}$, defined by a presentation over alphabet $E$, and denoted by $\mathsf{IG}(\mathcal{E})$. Recently, much effort has been put into investigating the structure of semigroups of the form $\mathsf{IG}(\mathcal{E})$, especially regarding their maximal subgroups. In this paper we take these investigations in a new direction by considering the word problem for $\mathsf{IG}(\mathcal{E})$. We prove two principal results, one positive and one negative. We show that, for a finite biordered set $\mathcal{E}$, it is decidable whether a given word $w \in E^*$ represents a regular element; if in addition one assumes that all maximal subgroups of $\mathsf{IG}(\mathcal{E})$ have decidable word problems, then the word problem in $\mathsf{IG}(\mathcal{E})$ restricted to regular words is decidable. On the other hand, we exhibit a biorder $\mathcal{E}$ arising from a finite idempotent semigroup $S$, such that the word problem for $\mathsf{IG}(\mathcal{E})$ is undecidable, even though all the maximal subgroups have decidable word problems. This is achieved by relating the word problem of $\mathsf{IG}(\mathcal{E})$ to the subgroup membership problem in finitely presented groups.