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.


  1. 1.
    Ahmed, C., Martin, P., Mazorchuk, V.: On the number of principal ideals in d-tonal partition monoids. arXiv:1503.06718
  2. 2.
    André, J., Araújo, J., Cameron, P.J.: The classification of partition homogeneous groups with applications to semigroup theory. J. Algebra 452, 288–310 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    André, J.M., Araújo, J., Konieczny, J.: Regular centralizers of idempotent transformations. Semigroup Forum 82(2), 307–318 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Araújo, J., Bentz, W., Konieczny, J.: The largest subsemilattices of the semigroup of endomorphisms of an independence algebra. Linear Algebra Appl. 458, 50–79 (2014)CrossRefzbMATHGoogle Scholar
  5. 5.
    Araújo, J., Bentz, W., Cameron, P.J., Royle, G., Schaefer, A.: Primitive groups and synchronization. Proc. Lond. Math. Soc. 113, 829–867 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Araújo, J., Bentz, W., Mitchell, J.D., Schneider, C.: The rank of the semigroup of transformations stabilising a partition of a finite set. Math. Proc. Camb. Philos. Soc. 159(2), 339–353 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Araújo, J., Bentz, W., Dobson, E., Konieczny, J., Morris, J.: Automorphism groups of circulant digraphs with applications to semigroup theory. Combinatorica 38, 1–28 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Araújo, J., Cameron, P.J.: Primitive groups synchronize non-uniform maps of extreme ranks. J. Comb. Theory Ser. B 106, 98–114 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Araújo, J., Cameron, P.J.: Two generalizations of homogeneity in groups with applications to regular semigroups. Trans. Am. Math. Soc. 368, 1159–1188 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Araújo, J., Cameron, P.J., Mitchell, J.D., Neuhoffer, M.: The classification of normalizing groups. J. Algebra 373, 481–490 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Araújo, J., Cameron, P.J., Steinberg, B.: Between primitive and 2-transitive: synchronization and its friends. Eur. Math. Soci. Surv. Math. Sci. 4(2), 101–184 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Araújo, J., Fountain, J.: The origins of independence algebras. Semigroups Lang. 54–67 (2004)Google Scholar
  13. 13.
    Araújo, J., Konieczny, J.: Semigroups of transformations preserving an equivalence relation and a cross-section. Commun. Algebra 32, 1917–1935 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Araújo, J., Konieczny, J.: Centralizers in the full transformation semigroup. Semigroup Forum 86, 1–31 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Araújo, J., Silva, F.C.: Semigroups of linear endomorphisms closed under conjugation. Commun. Algebra 28(8), 3679–3689 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Araújo, J., Wehrung, F.: Embedding properties of endomorphism semigroups. Fundam. Math. 202, 125–146 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cameron, P.J., Szabó, C.: Independence algebras. J. Lond. Math. Soc. 61, 321–334 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    East, J.: Generators and relations for partition monoids and algebras. J. Algebra 339, 1–26 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    East, J.: On the singular part of the partition monoid. Int. J. Algebra Comput. 21(1–2), 147–178 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Dolinka, I., East, J., Evangelou, A., FitzGerald, D., Ham, N., Hyde, J., Loughlin, N.: Enumeration of idempotents in diagram semigroups and algebras. J. Comb. Theory Ser. A 131, 119–152 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    FitzGerald, D.G., Lau, K.W.: On the partition monoid and some related semigroups. Bull. Aust. Math. Soc. 83(2), 273–288 (2011)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Fountain, J., Gould, V.: Relatively free algebras with weak exchange properties. J. Aust. Math. Soc. 75, 355–384 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Fountain, J., Gould, V.: Endomorphisms of relatively free algebras with weak exchange properties. Algebra Univ. 51, 257–285 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ganyushkin, O., Mazorchuk, V.: Classical Finite Transformation Semigroups. An Introduction. Algebra and Applications, vol. 9. Springer, London (2009)zbMATHGoogle Scholar
  25. 25.
    Gould, V.: Independence algebras. Algebra Univ. 33, 294–318 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Howie, J.M.: Fundamentals of Semigroup Theory, London Mathematical Society Monographs. New Series, vol. 12. The Clarendon Press, New York (1995)Google Scholar
  27. 27.
    Kudryavtseva, A., Mazorchuk, V.: Square matrices as a semigroup. research/pub/Mazorchuk9.pdf, July 6 (2015)
  28. 28.
    Levi, I.: Automorphisms of normal transformation semigroups. Proc. Edinb. Math. Soc. 28, 185–205 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Levi, I.: Automorphisms of normal partial transformation semigroups. Glasg. Math. J. 29, 149–157 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Levi, I.: Congruences on normal transformation semigroups. Math. Jpn. 52(2), 247–261 (2000)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Levi, I., McAlister, D.B., McFadden, R.B.: Groups associated with finite transformation semigroups. Semigroup Forum 61(3), 453–467 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Liber, A.: On symmetric generalized groups. Mat. Sb. N.S. 33(75), 531–544 (1953)MathSciNetGoogle Scholar
  33. 33.
    Mal’cev, A.: Symmetric groupoids. Mat. Sb. N. S. 31(73), 136–151 (1952)MathSciNetGoogle Scholar
  34. 34.
    Mal’cev, A.: Multiplicative congruences of matrices. Dokl. Akad. N. S. 90, 333–335 (1953)MathSciNetGoogle Scholar
  35. 35.
    McAlister, Donald B.: Semigroups generated by a group and an idempotent. Commun. Algebra 26(2), 515–547 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Neumann, P.M.: Primitive permutation groups and their section-regular partitions. Mich. Math. J. 58, 309–322 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Schein, B., Teclezghi, B.: Endomorphisms of finite full transformation semigroups. Proc. Am. Math. Soc. 126(9), 2579–2587 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Šutov, E.: Homomorphisms of the semigroup of all partial transformations. Izv. Vysshikh Uchebnykh Zaved. Mat. 22(3), 177–184 (1961)MathSciNetGoogle Scholar
  39. 39.
    Symons, J.S.V.: Normal transformation semigroups. J. Aust. Math. Soc. Ser. A 22(4), 385–390 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Urbanik, K.: A representation theorem for \(v^*\)-algebras. Fundam. Math. 52, 291–317 (1963)CrossRefzbMATHGoogle Scholar

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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