Semigroups of rectangular matrices under a sandwich operation

Let Mmn =Mmn(F) denote the set of all m× n matrices over a field F, and fix some n×m matrix A ∈ Mnm. An associative operation ? may be defined on Mmn by X ? Y = XAY for all X,Y ∈ Mmn, and the resulting sandwich semigroup is denoted Mmn = Mmn(F). In this article, we study Mmn as well as its subsemigroups Reg(Mmn) and EA mn (consisting of all regular elements and products of idempotents, respectively), as well as the ideals of Reg(Mmn). 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 Mmn(F1) and Mkl(F2). 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 note that all our results have applications to the variants Mn of the full linear monoid Mn = Mnn (in the case m = n), and to certain semigroups of linear transformations of restricted range or kernel (in the case that rank(A) is equal to one of m,n).