English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

With each semigroup one can associate a partial algebra, called the biorderedset, which captures important algebraic and geometric features of the structureof idempotents of that semigroup. For a biordered set $\mathcal{E}$, one canconstruct the free idempotent-generated semigroup over $\mathcal{E}$,$\mathsf{IG}(\mathcal{E})$, which is the free-est semigroup (in a definitecategorical sense) whose biorder of idempotents is isomorphic to $\mathcal{E}$.Studies of these intriguing objects have been recently focusing on theirparticular 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 thestructure of the free idempotent-generated semigroup over the biorder of$\mathcal{T}_n$, the full transformation monoid over an $n$-element set, mightbe worth pursuing. In 2019, together with Gould and Yang, the present authorshowed that the word problem for $\mathsf{IG}(\mathcal{E}_{\mathcal{T}_n})$ isalgorithmically soluble. In a recent work by the author, it was showed that,for a wide class of biorders $\mathcal{E}$, the algorithmic solution of theword problem revolves around the so-called vertex groups, which arise ascertain subgroups of direct products of pairs of maximal subgroups of$\mathsf{IG}(\mathcal{E})$. In this paper we determine these vertex groups forthe case when $\mathcal{E}$ is the biorder of idempotents of $\mathcal{T}_n$.

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