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Congruences on direct products of transformation and matrix monoids

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


João Araújo

Wolfram Bentz

Gracinda M. S. Gomes


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.


Araújo, J., Bentz, W., & Gomes, G. M. S. (2018). Congruences on direct products of transformation and matrix monoids. Semigroup Forum, 97(3), 384–416.

Journal Article Type Article
Acceptance Date Mar 5, 2018
Online Publication Date Apr 2, 2018
Publication Date 2018-12
Deposit Date Mar 18, 2018
Publicly Available Date Apr 3, 2019
Print ISSN 0037-1912
Electronic ISSN 1432-2137
Publisher Springer Verlag
Peer Reviewed Peer Reviewed
Volume 97
Issue 3
Pages 384–416
Keywords Algebra and Number Theory
Public URL
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