Elaborating the word problem for free idempotent-generated semigroups over the full transformation monoid

TL;DR

This paper investigates the structure of free idempotent-generated semigroups over the biorder of the full transformation monoid, focusing on the vertex groups that influence the word problem's algorithmic solution.

Contribution

It explicitly determines the vertex groups for the biorder of idempotents in the full transformation monoid, advancing understanding of their algebraic structure.

Findings

Vertex groups are explicitly characterized for the biorder of $ ext{T}_n$.

The results facilitate solving the word problem for these semigroups.

Provides a detailed algebraic description of the vertex groups involved.

Abstract

With each semigroup one can associate a partial algebra, called the biordered set, which captures important algebraic and geometric features of the structure of idempotents of that semigroup. For a biordered set $\mathcal{E}$, one can construct the free idempotent-generated semigroup over $\mathcal{E}$, $\mathsf{IG}(\mathcal{E})$, which is the free-est semigroup (in a definite categorical sense) whose biorder of idempotents is isomorphic to $\mathcal{E}$. Studies of these intriguing objects have been recently focusing on their particular aspects, such as maximal subgroups, the word problem, etc. In 2012, Gray and Ru\v{s}kuc pointed out that a more detailed investigation into the structure of the free idempotent-generated semigroup over the biorder of $\mathcal{T}_n$, the full transformation monoid over an $n$-element set, might be worth pursuing. In 2019, together with Gould and Yang, the present author showed that the word problem for $\mathsf{IG}(\mathcal{E}_{\mathcal{T}_n})$ is algorithmically soluble. In a recent work by the author, it was showed that, for a wide class of biorders $\mathcal{E}$, the algorithmic solution of the word problem revolves around the so-called vertex groups, which arise as certain subgroups of direct products of pairs of maximal subgroups of $\mathsf{IG}(\mathcal{E})$. In this paper we determine these vertex groups for the case when $\mathcal{E}$ is the biorder of idempotents of $\mathcal{T}_n$.