Congruences on direct products of transformation and matrix monoids

  • João Araújo
  • Wolfram Bentz
  • Gracinda M. S. Gomes
Research Article


Mal\('\)cev described the congruences of the monoid \(\mathcal {T}_n\) of all full transformations on a finite set \(X_n=\{1, \dots ,n\}\). Since then, congruences have been characterized in various other monoids of (partial) transformations on \(X_n\), such as the symmetric inverse monoid \(\mathcal {I}_n\) of all injective partial transformations, or the monoid \(\mathcal {PT}_n\) of all partial transformations. The first aim of this paper is to describe the congruences of the direct products \(Q_m\times P_n\), where Q and P belong to \(\{\mathcal {T}, \mathcal {PT},\mathcal {I}\}\). Mal\('\)cev also provided a similar description of the congruences on the multiplicative monoid \(F_n\) of all \(n\times n\) matrices with entries in a field F; our second aim is to provide a description of the principal congruences of \(F_m \times F_n\). The paper finishes with some comments on the congruences of products of more than two transformation semigroups, and on a number of related open problems.


Monoid Congruences Green relations 



The authors were supported by FCT (Portugal) through project UID/MULTI/04621/2013 of CEMAT-Ciências. Wolfram Bentz has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under Grant Agreement No. PCOFUND-GA-2009-246542 and from the Foundation for Science and Technology of Portugal under PCOFUND-GA-2009-246542 and SFRH/BCC/52684/2014. The authors wish to thank the referee for his or her helpful remarks.


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Copyright information

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Authors and Affiliations

  • João Araújo
    • 1
    • 2
  • Wolfram Bentz
    • 3
  • Gracinda M. S. Gomes
    • 2
  1. 1.Universidade AbertaLisbonPortugal
  2. 2.Departamento de Matemática, CEMAT-Ciências, Faculdade de CiênciasUniversidade de LisboaLisbonPortugal
  3. 3.School of Mathematics and Physical SciencesUniversity of HullHullUK

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