Advertisement

Semigroup Forum

, Volume 97, Issue 2, pp 325–352 | Cite as

The Nine Lemma and the push forward construction for special Schreier extensions of monoids with operations

  • Nelson Martins-Ferreira
  • Andrea Montoli
  • Manuela Sobral
Research Article
  • 27 Downloads

Abstract

We show that the Nine Lemma holds for special Schreier extensions of monoids with operations. This fact is used to obtain a push forward construction for special Schreier extensions with abelian kernel. This construction permits to give a functorial description of the Baer sum of such extensions.

Keywords

Monoids with operations Special Schreier extension Nine Lemma Push forward Baer sum 

Notes

Acknowledgements

We wish to express our gratitude to Alex Patchkoria for pointing out to us the existence of some old literature, of not easy access, related to the subject of this paper. This work was partially supported by the Centre for Mathematics of the University of Coimbra – UID/MAT/00324/2013, by ESTG and CDRSP from the Polytechnical Institute of Leiria – UID/Multi/04044/2013, funded by the Portuguese Government through FCT/MCTES and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020. This work was partially supported by the Programma per Giovani Ricercatori “Rita Levi-Montalcini”, Funded by the Italian government through MIUR.

References

  1. 1.
    Barr, M.: Exact Categories, Lecture Notes in Mathematics, vol. 236, pp. 1–120. Springer, Berlin (1971)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, vol. 45 (2013)Google Scholar
  3. 3.
    Bourn, D., Martins-Ferreira, N., Montoli, A., Sobral, M.: Schreier split epimorphisms between monoids. Semigroup Forum 88, 739–752 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bourn, D., Martins-Ferreira, N., Montoli, A., Sobral, M.: Monoids and pointed \(S\)-protomodular categories. Homol. Homotopy Appl. 18(1), 151–172 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Grillet, P.A.: Left coset extensions. Semigroup Forum 7, 200–263 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hoff, G.: On the cohomology of categories. Rend. Mate. 7, 169–192 (1974)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Hoff, G.: Cohomologies et extensions de catégories. Math. Scand. 74, 191–207 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Leech, J.: Extending groups by monoids. J. Algebra 74, 1–19 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Mal’cev, A.I.: On the general theory of algebraic systems. Mat. Sbornik N.S. 35, 3–20 (1954)Google Scholar
  10. 10.
    Martins-Ferreira, N., Montoli, A.: On the “Smith is Huq” condition in \(S\)-protomodular categories. Appl. Categ. Struct. 25, 59–75 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Martins-Ferreira, N., Montoli, A., Sobral, M.: Semidirect products and crossed modules in monoids with operations. J. Pure Appl. Algebra 217, 334–347 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Martins-Ferreira, N., Montoli, A., Sobral, M.: Baer sums of special Schreier extensions of monoids. Semigroup Forum 93, 403–415 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Nguen Suan Tuen: Extensions of groups and monoids. Sakharth. SSR Mecn. Akad. Moambe 83(1), 25–28 (1976)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Orzech, G.: Obstruction theory in algebraic categories, I. J. Pure Appl. Algebra 2, 287–314 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Patchkoria, A.: Extensions of semimodules by monoids and their cohomological characterization. Bull. Georgian Acad. Sci. 86, 21–24 (1977)Google Scholar
  16. 16.
    Patchkoria, A.: Cohomology of monoids with coefficients in semimodules. Bull. Georgian Acad. Sci. 86, 545–548 (1977)MathSciNetGoogle Scholar
  17. 17.
    Patchkoria, A.: Crossed semimodules and Schreier internal categories in the category of monoids. Georgian Math. J. 5(6), 575–581 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Patchkoria, A.: Cohomology monoids of monoids with coefficients in semimodules I. J. Homotopy Relat. Struct. 9, 239255 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Patchkoria, A.: Cohomology monoids of monoids with coefficients in semimodules II. Semigroup Forum 97, 131–153 (2018)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Porter, T.: Extensions, crossed modules and internal categories in categories of groups with operations. Proce. Edinb. Math. Soc. 30, 373–381 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rédei, L.: Die Verallgemeinerung der Schreierischen Erweiterungstheorie. Acta Sci. Math. Szeged 14, 252–273 (1952)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Wells, C.: Extension theory for monoids. Semigroup Forum 16, 13–35 (1978)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Nelson Martins-Ferreira
    • 1
  • Andrea Montoli
    • 2
  • Manuela Sobral
    • 3
  1. 1.ESTG, CDRSPInstituto Politécnico de LeiriaLeiriaPortugal
  2. 2.Dipartimento di Matematica “Federigo Enriques”Università degli Studi di MilanoMilanItaly
  3. 3.CMUC and Departamento de MatemáticaUniversidade de CoimbraCoimbraPortugal

Personalised recommendations