Australian Category Seminar

The partition monoid $\mathcal{P}_n$ provides a unifying algebraic structure for a variety of notable submonoids — collectively referred to as diagram monoids — that have intuitive graphical representations. These include the Brauer monoid, the full transformation monoid, and the dual symmetric inverse monoid $\mathcal{J}_n$ (and, by extension, the Temperley-Lieb monoid, the symmetric inverse monoid $\mathcal{I}_n$ , and the symmetric group). The partition monoid is a DR-semigroup with respect to $E(\mathcal{J}_n)$ and $E(\mathcal{I}_n)$ (the semilattices of $\mathcal{J}_n$ and $\mathcal{I}_n$), meaning it is equipped with two sets of domain and range unary operations whose image are $E(\mathcal{J}_n)$ and $E(\mathcal{I}_n)$ and that fulfil certain axioms which have natural categorical interpretations. Recent work by East and Gray studying these operations led to the discovery of two new diagram monoids: the full-domain partition monoid $\mathcal{P}_n^{\mathrm{fd}}$ and the trivial-cokernel partition monoid $\mathcal{P}_n^{\mathrm{tc}}$. The first of these is a right-restriction semigroup (and hence also Ehresmann) with respect to $E(\mathcal{J}_n)$, whilst the second is a DR-semigroup with respect to $E(\mathcal{I}_n)$. In this talk, I will discuss the monoids $\mathcal{P}_n^{\mathrm{fd}}$, $\mathcal{P}_n^{\mathrm{tc}}$, and their domain and range operations, as well as recent work that was undertaken in collaboration with James East, Matthias Fresacher, Mengfan Lyu, and Azeef Muhammed to find presentations for the two monoids. I will also share some novel examples of left- and right-regular bands that were discovered whilst proving these presentations.