Semigroup Forum

, Volume 88, Issue 3, pp 739–752 | Cite as

Schreier split epimorphisms between monoids

  • Dominique Bourn
  • Nelson Martins-Ferreira
  • Andrea Montoli
  • Manuela Sobral
Research Article


We explore some properties of Schreier split epimorphisms between monoids, which correspond to monoid actions. In particular, we prove that the split short five lemma holds for monoids, when it is restricted to Schreier split epimorphisms, and that any Schreier reflexive relation is transitive, partially recovering in monoids a classical property of Mal’tsev varieties.


Schreier split epimorphisms Monoids Split short five lemma Internal relations 



This work was partially supported by the Centro de Matemática da Universidade de Coimbra (CMUC), funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT - Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0324/2013 and grants number PTDC/MAT/120222/2010 and SFRH/BPD/69661/2010, and also by ESTG and CDRSP from the Polytechnical Institute of Leiria.


  1. 1.
    Bourn, D.: Normalization equivalence, kernel equivalence and affine categories. Lecture Notes in Mathematics, pp. 43–62. Springer-Verlag, Berlin (1991)Google Scholar
  2. 2.
    Bourn, D., Martins-Ferreira, N., Montoli, A., Sobral, M.: Schreier split epimorphisms in monoids and in semirings, Textos de Matemática Série B. Departamento de Matemática da Universidade de Coimbra, Coimbra (2014)Google Scholar
  3. 3.
    Carboni, A., Lambek, J., Pedicchio, M.C.: Diagram chasing in Mal’cev categories. J. Pure Appl. Algebra 69, 271–284 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Lavendhomme, R., Roisin, J.R.: Cohomologie non abélienne de structures algébriques. J. Algebra 67, 385–414 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Mal’cev, A.I.: On the general theory of algebraic systems. Mat. Sbornik N. S. 35, 3–20 (1954)MathSciNetGoogle Scholar
  6. 6.
    Martins-Ferreira, N., Montoli, A., Sobral, M.: Semidirect products and crossed modules in monoids with operations. J. Pure Appl. Algebra 217, 334–347 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Patchkoria, A.: Crossed semimodules and Schreier internal categories in the category of monoids. Georgian Math. J. 5(6), 575–581 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Rédei, L.: Die Verallgemeinerung der Schreierschen Erweiterungstheorie. Acta Sci. Math. (Szeged) 14, 252–273 (1952)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Dominique Bourn
    • 1
  • Nelson Martins-Ferreira
    • 2
  • Andrea Montoli
    • 3
  • Manuela Sobral
    • 4
  1. 1.Laboratoire de Mathématiques pures et appliquéesUniversité du Littoral Côte d’OpaleCalaisFrance
  2. 2.ESTG, CDRSPInstituto Politécnico de LeiriaLeiriaPortugal
  3. 3.CMUC, University of CoimbraCoimbraPortugal
  4. 4.CMUC and Department of MathematicsUniversity of CoimbraCoimbraPortugal

Personalised recommendations