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Congruences on direct products of transformation and matrix monoids

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Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. Universidade Aberta, R. Escola Politécnica, 147, 1269-001, Lisbon, Portugal

    João Araújo

  2. Departamento de Matemática, CEMAT-Ciências, Faculdade de Ciências, Universidade de Lisboa, 1749-016, Lisbon, Portugal

    João Araújo & Gracinda M. S. Gomes

  3. School of Mathematics and Physical Sciences, University of Hull, Hull, HU6 7RX, UK

    Wolfram Bentz

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  1. João Araújo
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Correspondence to Wolfram Bentz.

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Communicated by Mikhail Volkov.

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Araújo, J., Bentz, W. & Gomes, G.M.S. Congruences on direct products of transformation and matrix monoids. Semigroup Forum 97, 384–416 (2018). https://doi.org/10.1007/s00233-018-9931-8

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