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Congruences on direct products of transformation and matrix monoids

Araújo, João; Bentz, Wolfram; Gomes, Gracinda M. S.

Authors

João Araújo

Gracinda M. S. Gomes



Abstract

Malcev described the congruences of the monoid Tn of all full transformations on a finite set Xn={1,…,n}. Since then, congruences have been characterized in various other monoids of (partial) transformations on Xn, such as the symmetric inverse monoid Inn of all injective partial transformations, or the monoid PTn of all partial transformations. The first aim of this paper is to describe the congruences of the direct products Qm×Pn, where Q and P belong to {T,PT,In}. Malcev also provided a similar description of the congruences on the multiplicative monoid Fn of all n×n matrices with entries in a field F, our second aim is provide a description of the principal congruences of Fm×Fn. The paper finishes with some comments on the congruences of products of more than two transformation semigroups, and a fairly large number of open problems.

Journal Article Type Article
Publication Date 2018-12
Print ISSN 0037-1912
Electronic ISSN 1432-2137
Publisher Springer Verlag
Peer Reviewed Peer Reviewed
Volume 97
Issue 3
Pages 384–416
Institution Citation Araújo, J., Bentz, W., & Gomes, G. M. S. (2018). Congruences on direct products of transformation and matrix monoids. Semigroup Forum, 97(3), 384–416. https://doi.org/10.1007/s00233-018-9931-8
DOI https://doi.org/10.1007/s00233-018-9931-8
Keywords Algebra and Number Theory
Publisher URL https://link.springer.com/article/10.1007%2Fs00233-018-9931-8
Copyright Statement ©2018 University of Hull

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