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

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 $$\mathscr {E}$$ , one can construct the free idempotent-generated semigroup over $$\mathscr {E}$$ , $$\mathsf {IG}(\mathscr {E})$$ , which is the free-est semigroup (in a definite categorical sense) whose biorder of idempotents is isomorphic to $$\mathscr {E}$$ . 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 Ruškuc pointed out that a more detailed investigation into the structure of the free idempotent-generated semigroup over the biorder of $$\mathscr {T}_n$$ , 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 $$\mathsf {IG}(\mathscr {E}_{\mathscr {T}_n})$$ is algorithmically soluble. In a recent work by the author, it was showed that, for a wide class of biorders $$\mathscr {E}$$ , 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 $$\mathsf {IG}(\mathscr {E})$$ . In this paper we determine these vertex groups for the case when $$\mathscr {E}$$ is the biorder of idempotents of $$\mathscr {T}_n$$ .