Semigroups of rectangular matrices under a sandwich operation

Abstract

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 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. We study ${\mathcal {M}}_{mn}^A$ as well as its subsemigroups $\hbox {Reg}({\mathcal {M}}_{mn}^A)$ and ${\mathcal {E}}_{mn}^A$ (consisting of all regular elements and products of idempotents, respectively), and the ideals of $\hbox {Reg}({\mathcal {M}}_{mn}^A)$. 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 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 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 $\hbox {rank}(A)$ is equal to one of m, n).