A group-theoretical interpretation of the word problem for free idempotent generated semigroups

TL;DR

This paper explores the word problem in free idempotent generated semigroups, linking it to constraint satisfaction problems and establishing decidability results for certain classes.

Contribution

It introduces a group-theoretical framework for the word problem in free idempotent generated semigroups with finite biordered sets, connecting it to rational subset problems.

Findings

Word problem equivalent to constraint satisfaction problems.

Decidability established for specific classes of semigroups.

Finite biordered sets lead to weakly abundant semigroups satisfying the congruence condition.

Abstract

The set of idempotents of any semigroup carries the structure of a biordered set, which contains a great deal of information concerning the idempotent generated subsemigroup of the semigroup in question. This leads to the construction of a free idempotent generated semigroup $\mathsf{IG}(\mathcal{E})$ - the `free-est' semigroup with a given biordered set $\mathcal{E}$ of idempotents. We show that when $\mathcal{E}$ is finite, the word problem for $\mathsf{IG}(\mathcal{E})$ is equivalent to a family of constraint satisfaction problems involving rational subsets of direct products of pairs of maximal subgroups of $\mathsf{IG}(\mathcal{E})$. As an application, we obtain decidability of the word problem for an important class of examples. Also, we prove that for finite $\mathcal{E}$, $\mathsf{IG}(\mathcal{E})$ is always a weakly abundant semigroup satisfying the congruence condition.