Congruences on direct products of transformation and matrix monoids


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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14


  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)

    MathSciNet  Article  Google Scholar 

  3. 3.

    André, J.M., Araújo, J., Konieczny, J.: Regular centralizers of idempotent transformations. Semigroup Forum 82(2), 307–318 (2011)

    MathSciNet  Article  Google 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)

    Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Araújo, J., Cameron, P.J., Mitchell, J.D., Neuhoffer, M.: The classification of normalizing groups. J. Algebra 373, 481–490 (2013)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Araújo, J., Fountain, J.: The origins of independence algebras. Semigroups Lang. 54–67 (2004)

  13. 13.

    Araújo, J., Konieczny, J.: Semigroups of transformations preserving an equivalence relation and a cross-section. Commun. Algebra 32, 1917–1935 (2004)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Araújo, J., Konieczny, J.: Centralizers in the full transformation semigroup. Semigroup Forum 86, 1–31 (2013)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Araújo, J., Silva, F.C.: Semigroups of linear endomorphisms closed under conjugation. Commun. Algebra 28(8), 3679–3689 (2000)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Araújo, J., Wehrung, F.: Embedding properties of endomorphism semigroups. Fundam. Math. 202, 125–146 (2009)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Cameron, P.J., Szabó, C.: Independence algebras. J. Lond. Math. Soc. 61, 321–334 (2000)

    MathSciNet  Article  Google Scholar 

  18. 18.

    East, J.: Generators and relations for partition monoids and algebras. J. Algebra 339, 1–26 (2011)

    MathSciNet  Article  Google Scholar 

  19. 19.

    East, J.: On the singular part of the partition monoid. Int. J. Algebra Comput. 21(1–2), 147–178 (2011)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Fountain, J., Gould, V.: Relatively free algebras with weak exchange properties. J. Aust. Math. Soc. 75, 355–384 (2003)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Fountain, J., Gould, V.: Endomorphisms of relatively free algebras with weak exchange properties. Algebra Univ. 51, 257–285 (2004)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Ganyushkin, O., Mazorchuk, V.: Classical Finite Transformation Semigroups. An Introduction. Algebra and Applications, vol. 9. Springer, London (2009)

    Google Scholar 

  25. 25.

    Gould, V.: Independence algebras. Algebra Univ. 33, 294–318 (1995)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Levi, I.: Automorphisms of normal partial transformation semigroups. Glasg. Math. J. 29, 149–157 (1987)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Levi, I.: Congruences on normal transformation semigroups. Math. Jpn. 52(2), 247–261 (2000)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Levi, I., McAlister, D.B., McFadden, R.B.: Groups associated with finite transformation semigroups. Semigroup Forum 61(3), 453–467 (2000)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Liber, A.: On symmetric generalized groups. Mat. Sb. N.S. 33(75), 531–544 (1953)

    MathSciNet  Google Scholar 

  33. 33.

    Mal’cev, A.: Symmetric groupoids. Mat. Sb. N. S. 31(73), 136–151 (1952)

    MathSciNet  Google Scholar 

  34. 34.

    Mal’cev, A.: Multiplicative congruences of matrices. Dokl. Akad. N. S. 90, 333–335 (1953)

    MathSciNet  Google Scholar 

  35. 35.

    McAlister, Donald B.: Semigroups generated by a group and an idempotent. Commun. Algebra 26(2), 515–547 (1998)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Neumann, P.M.: Primitive permutation groups and their section-regular partitions. Mich. Math. J. 58, 309–322 (2009)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Schein, B., Teclezghi, B.: Endomorphisms of finite full transformation semigroups. Proc. Am. Math. Soc. 126(9), 2579–2587 (1998)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Šutov, E.: Homomorphisms of the semigroup of all partial transformations. Izv. Vysshikh Uchebnykh Zaved. Mat. 22(3), 177–184 (1961)

    MathSciNet  Google Scholar 

  39. 39.

    Symons, J.S.V.: Normal transformation semigroups. J. Aust. Math. Soc. Ser. A 22(4), 385–390 (1976)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Urbanik, K.: A representation theorem for \(v^*\)-algebras. Fundam. Math. 52, 291–317 (1963)

    MathSciNet  Article  Google Scholar 

Download references


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.

Author information



Corresponding author

Correspondence to Wolfram Bentz.

Additional information

Communicated by Mikhail Volkov.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation


  • Monoid
  • Congruences
  • Green relations