Semigroups of rectangular matrices under a sandwich operation

Abstract

© 2017, Springer Science+Business Media New York. 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 MmnA=MmnA(F). These semigroups are closely related to Munn rings, which are fundamental tools in the representation theory of finite semigroups. We study MmnA as well as its subsemigroups Reg(MmnA) and EmnA (consisting of all regular elements and products of idempotents, respectively), and the ideals of Reg(MmnA). 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 MmnA(F1) and MklB(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 hope this framework will be useful in studies of sandwich semigroups in other categories. We note that all our results have applications to the variantsMnA of the full linear monoid Mn (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).