Semigroups of rectangular matrices under a sandwich operation

Let $\mathcal M_{mn}=\mathcal M_{mn}(\mathbb F)$ denote the set of all $m\times n$ matrices over a field $\mathbb F$, and fix some $n\times m$ matrix $A\in\mathcal M_{nm}$. An associative operation $\star$ may be defined on $\mathcal M_{mn}$ by $X\star Y=XAY$ for all $X,Y\in\mathcal M_{mn}$, and the resulting \emph{sandwich semigroup} is denoted $\mathcal M_{mn}^A=\mathcal M_{mn}^A(\mathbb F)$. These semigroups are closely related to Munn rings, which are fundamental tools in the representation theory of finite semigroups. In this article, we study $\mathcal M_{mn}^A$ as well as its subsemigroups $\operatorname{Reg}(\mathcal M_{mn}^A)$ and $\mathcal E_{mn}^A$ (consisting of all regular elements and products of idempotents, respectively), as well as the ideals of $\operatorname{Reg}(\mathcal M_{mn}^A)$. Among other results, we: characterise the regular elements, determine Green's relations and preorders, calculate the minimal number of matrices (or idempotent matrices, if applicable) required to generate each semigroup we consider, and classify the isomorphisms between finite sandwich semigroups $\mathcal M_{mn}^A(\mathbb F_1)$ and $\mathcal M_{kl}^B(\mathbb F_2)$. Along the way, we develop a general theory of sandwich semigroups in a suitably defined class of \emph{partial semigroups} related to Ehresmann-style "arrows only" categories, we hope this framework will be useful in studies of sandwich semigroups in other categories. We note that all our results have applications to the \emph{variants} $\mathcal M_n^A$ of the full linear monoid $\mathcal M_n$ (in the case $m=n$), and to certain semigroups of linear transformations of restricted range or kernel (in the case that $\operatorname{rank}(A)$ is equal to one of $m,n$).