Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Elaborating the word problem for free idempotent-generated semigroups
  over the full transformation monoid   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 $\mathcal{T}_n$.

31
The categories of this paper are: ao-sci
6
The categories of this paper are: astro-ph
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
14
The categories of this paper are: math-ph
36
The categories of this paper are: cmp-lg
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
18
The categories of this paper are: econ
28
The categories of this paper are: alg-geom
23
The categories of this paper are: solv-int
3
The categories of this paper are: physics
13
The categories of this paper are: quant-ph
33
The categories of this paper are: plasm-ph
8
The categories of this paper are: hep-ex
21
The categories of this paper are: adap-org