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Formations of orthodox semigroups

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Abstract

Taking formations of groups and of inverse semigroups as the starting point, formations of orthodox semigroups are defined, as well as the wider class of i-formations (i standing for idempotent-separating). The relation between the nature of a class of inverse semigroups \(\mathscr {F}\) [of groups \(\mathscr {G}\)] and that of certain classes of orthodox semigroups with associated inverse semigroups in \(\mathscr {F}\) [groups in \(\mathscr {G}\)] is discussed. The product of formations of orthodox semigroups, in particular of R-unipotent semigroups, is considered, and a product like the Gaschütz product known for groups is presented for i-formations. The paper concludes with a list of questions.

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References

  1. Almeida, J.: Finite Semigroups and Universal Algebra. Series in Algebra, vol. 3. World Scientific, Singapore (1994)

    Google Scholar 

  2. Ballester-Bolinches, A., Ezquerro, L.M.: Classes of Finite Groups, Mathematics and Its Applications, vol. 584. Springer, Dordrecht (2006)

    Google Scholar 

  3. Ballester-Bolinches, A., Pin, J.-E., Soler-Escrivà, X.: Formations of finite monoids and formal languages: Eilenberg’s variety theorem revisited. Forum Math. 26, 1737–1761 (2014)

    Article  MathSciNet  Google Scholar 

  4. Branco, M.J., Gomes, G.M., Pin, J., Soler-Escrivà, X.: On formations of monoids. J. Pure Appl. Algebra 224(11), 106401 (2020)

  5. Doerk, K., Hawkes, T.: Finite Soluble Groups. Walter de Gruyter, Berlin (1992)

    Book  Google Scholar 

  6. Gaschütz, W.: Zur Theorie der endlichen auflösbaren Gruppen. Math. Z. 80(1), 300–305 (1962)

    Article  MathSciNet  Google Scholar 

  7. Gomes, G.M.S.: A characterization of the group congruences on a semigroup. Semigroup Forum 46(1), 48–53 (1993)

    Article  MathSciNet  Google Scholar 

  8. Gomes, G.M.S.: Orthodox congruences on regular semigroups. Semigroup Forum 37, 149–166 (1998)

    Article  MathSciNet  Google Scholar 

  9. Gomes, G.M.S., Nobre, I.J.: Formations of inverse semigroups. Semigroup Forum 105, 1–27 (2022)

    Article  MathSciNet  Google Scholar 

  10. Gusev, S.V., Lee, E.W.H., Vernikov, B.M.: The lattice of varieties of monoids. Jpn. J. Math. 17, 117–183 (2022)

    Article  MathSciNet  Google Scholar 

  11. Hall, T.E.: Congruences and Green’s relations on regular semigroups. Glasgow Math. J. 13(2), 167–175 (1972)

    Article  MathSciNet  Google Scholar 

  12. Hall, T.E.: Identities for existence varieties of regular semigroups. Bull. Austral. Math. Soc. 40(1), 59–77 (1989)

    Article  MathSciNet  Google Scholar 

  13. Howie, J.M.: An Introduction to Semigroup Theory. London Mathematical Society Monographs, vol. 7. Academic Press, London (1976)

    Google Scholar 

  14. Howie, J.M.: Fundamentals of Semigroup Theory, London Mathematical Society Monographs, New Series, vol. 12. Oxford University Press, Oxford (1995)

    Book  Google Scholar 

  15. Jones, P.R.: A basis theorem for free inverse semigroups. J. Algebra 49(1), 172–190 (1977)

    Article  MathSciNet  Google Scholar 

  16. Jones, P.R., Trotter, P.G.: Joins of inverse semigroup varieties and band varieties. Int. J. Algebra Comput. 1(3), 371–386 (1991)

    Article  MathSciNet  Google Scholar 

  17. Jones, P.R., Trotter, P.G.: Semidirect products of regular semigroups. Trans. Am. Math. Soc. 349(11), 4265–4310 (1997)

    Article  MathSciNet  Google Scholar 

  18. Kad’ourek, J., Szendrei, M.B.: A new approach in the theory of orthodox semigroups. Semigroup Forum 40(1), 257–296 (1990)

    Article  MathSciNet  Google Scholar 

  19. Kogalovskii, S.R.: On Birkhoff’s theorem. Uspekhi Matem. Nauk 20(5), 206–207 (1965) (In Russian)

  20. Mills, J.E.: Certain congruences on orthodox semigroups. Pac. J. Math. 64(1), 217–226 (1976)

    Article  MathSciNet  Google Scholar 

  21. Reilly, N.R.: Modular sublattices of the lattice of varieties of inverse semigroups. Pac. J. Math. 89(2), 405–417 (1980)

    Article  MathSciNet  Google Scholar 

  22. Shevrin, L.N., Vernikov, B.M., Volkov, M.V.: Lattices of semigroup varieties. Russ. Math. (Iz. VUZ) 53(1), 1–28 (2009)

    MathSciNet  Google Scholar 

  23. Wenbin, G.: The Theory of Classes of Groups, Mathematics and Its Applications, vol. 505. Springer, Dordrecht (2000)

    Book  Google Scholar 

  24. Wenbin, G., Shum, K.: Minimal formations of universal algebras. Discuss. Math. Gen. Algebra Appl. 21(2), 201–205 (2001)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was developed within the activities of Centro de Matemática Computacional e Estocástica, CEMAT, and Departamento de Matemática da Faculdade de Ciências da Universidade de Lisboa, within the projects UIDB/04621/2020 and UIDP/04621/2020, financed by Fundação para a Ciência e a Tecnologia, FCT. The authors would like to thank Adolfo Ballester-Bolinches and Mikhail Volkov for providing some of the references, Luís Sequeira for computing various examples and Mário J.J. Branco for some interesting discussions.

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Authors and Affiliations

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

    Gracinda M. S. Gomes & Ana-Catarina C. Monteiro

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  1. Gracinda M. S. Gomes
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  2. Ana-Catarina C. Monteiro
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Correspondence to Gracinda M. S. Gomes.

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Communicated by Victoria Gould.

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Gomes, G.M.S., Monteiro, AC.C. Formations of orthodox semigroups. Semigroup Forum 107, 651–679 (2023). https://doi.org/10.1007/s00233-023-10390-x

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