Semigroup Forum

, Volume 96, Issue 2, pp 253–300 | Cite as

Semigroups of rectangular matrices under a sandwich operation

Research Article
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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 mn).

Keywords

Matrix semigroups Sandwich semigroups Variants Idempotents Generators Rank Idempotent rank Munn rings Generalised matrix algebras 

Notes

Acknowledgements

The first named author gratefully acknowledges the support of Grant No. 174019 of the Ministry of Education, Science, and Technological Development of the Republic of Serbia. The authors wish to thank Dr. Attila Egri-Nagy and Dr. James Mitchell for constructing the GAP [67] code that enabled us to produce the egg-box diagrams from Figs. 3, 4, 5, 7 and 8.

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Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsUniversity of Novi SadNovi SadSerbia
  2. 2.Centre for Research in Mathematics, School of Computing, Engineering and MathematicsWestern Sydney UniversityPenrithAustralia

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