Advertisement

Algebra universalis

, 80:46 | Cite as

Bases for pseudovarieties closed under bideterministic product

  • Alfredo CostaEmail author
  • Ana Escada
Article
  • 45 Downloads

Abstract

We show that if \({\mathsf {V}}\) is a semigroup pseudovariety containing the finite semilattices and contained in \(\mathsf {DS}\), then it has a basis of pseudoidentities between finite products of regular pseudowords if, and only if, the corresponding variety of languages is closed under bideterministic product. The key to this equivalence is a weak generalization of the existence and uniqueness of \({\mathsf {J}}\)-reduced factorizations. This equational approach is used to address the locality of some pseudovarieties. In particular, it is shown that \(\mathsf {DH}\cap \mathsf {ECom}\) is local, for any group pseudovariety \({\mathsf {H}}\).

Keywords

Profinite semigroups Monoids Pseudovarieties Bideterministic product Basis of pseudoidentities Local pseudovarieties 

Mathematics Subject Classification

20M07 20M05 20M35 18B40 

Notes

References

  1. 1.
    Almeida, J.: Finite Semigroups and Universal Algebra. World Scientific, Singapore (1995). English translationGoogle Scholar
  2. 2.
    Almeida, J.: A syntactical proof of locality of DA. Int. J. Algebra Comput. 6, 165–177 (1996)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Almeida, J.: Profinite semigroups and applications. In: Kudryavtsev, V.B., Rosenberg, I.G. (eds.) Structural Theory of Automata, Semigroups and Universal Algebra, pp. 1–45. Springer, New York (2005)Google Scholar
  4. 4.
    Almeida, J., Azevedo, A.: On regular implicit operations. Portugal. Math. 50, 35–61 (1993)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Almeida, J., Azevedo, A., Teixeira, L.: On finitely based pseudovarieties of the forms V\(*\)D and V\(*\)D\(_{n}\). J. Pure Appl. Algebra 146, 1–15 (2000)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Almeida, J., Costa, A.: Infinite-vertex free profinite semigroupoids and symbolic dynamics. J. Pure Appl. Algebra 213, 605–631 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Almeida, J., Costa, A.: Equidivisible pseudovarieties of semigroups. Publ. Math. Debrecen 90, 435–453 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Almeida, J., Costa, A., Costa, J.C., Zeitoun, M.: The linear nature of pseudowords. Publ. Mat. 63, 361–422 (2019)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Almeida, J., Escada, A.: The globals of some subpseudovarieties of DA. Int. J. Algebra Comput. 14, 525–549 (2004)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Almeida, J., Weil, P.: Reduced factorizations in free profinite groups and join decompositions of pseudovarieties. Int. J. Algebra Comput. 4, 375–403 (1994)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Almeida, J., Weil, P.: Relatively free profinite monoids: an introduction and examples. In: Fountain, J.B. (ed.) Semigroups, Formal Languages and Groups, vol. 466, pp. 73–117. Kluwer, Dordrecht (1995)CrossRefGoogle Scholar
  12. 12.
    Almeida, J., Weil, P.: Profinite categories and semidirect products. J. Pure Appl. Algebra 123, 1–50 (1998)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Auinger, K.: Join decompositions of pseudovarieties involving semigroups with commuting idempotents. J. Pure Appl. Algebra 170, 115–129 (2002)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Azevedo, A.: Operações implícitas sobre pseudovariedades de semigrupos. Aplicações. Ph.D. thesis, Universidade do Porto (1989)Google Scholar
  15. 15.
    Branco, M.J.J.: On the Pin-Thérien expansion of idempotent monoids. Semigroup Forum 49, 329–334 (1994)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Branco, M.J.J.: Two algebraic approaches to variants of the concatenation product. Theor. Comput. Sci. 369, 406–426 (2006)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Branco, M.J.J.: Deterministic concatenation product of languages recognized by finite idempotent monoids. Semigroup Forum 74, 379–409 (2007)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Costa, A.: Pseudovarieties defining classes of sofic subshifts closed under taking shift equivalent subshifts. J. Pure Appl. Algebra 209, 517–530 (2007)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Costa, A., Escada, A.: Some operators that preserve the locality of a pseudovariety of semigroups. Int. J. Algebra Comput. 23, 583–610 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Eilenberg, S.: Automata, Languages and Machines, vol. B. Academic Press, New York (1976)zbMATHGoogle Scholar
  21. 21.
    Jiang, Z.: On the Pin-Thérien expansion of completely regular monoids. Semigroup Forum 60, 1–3 (2000)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Jones, P.R.: Monoid varieties defined by \(x^{n+1}=x\) are local. Semigroup Forum 47, 318–326 (1993)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Jones, P.R.: Profinite categories, implicit operations and pseudovarieties of categories. J. Pure Appl. Algebra 109, 61–95 (1996)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Jones, P.R., Szendrei, M.B.: Local varieties of completely regular monoids. J. Algebra 150, 1–27 (1992)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Kaďourek, J.: On the locality of the pseudovariety DG. J. Inst. Math. Jussieu 7, 93–180 (2008)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Pin, J.E.: Propriétés syntactiques du produit non ambigu. In: Automata, languages and programming (Proc. Seventh Internat. Colloq., Noordwijkerhout, 1980), Lect. Notes in Comput. Sci., vol. 85, pp. 483–499. Springer, Berlin (1980)Google Scholar
  27. 27.
    Pin, J.E.: Varieties of Formal Languages. Plenum, London (1986). English translationCrossRefGoogle Scholar
  28. 28.
    Pin, J.E.: Syntactic semigroups, chap. 10. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages. Springer, New York (1997)Google Scholar
  29. 29.
    Pin, J.E., Straubing, H., Thérien, D.: Locally trivial categories and unambiguous concatenation. J. Pure Appl. Algebra 52, 297–311 (1988)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Pin, J.E., Thérien, D.: The bideterministic concatenation product. Int. J. Algebra Comput. 3, 535–555 (1993)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Pin, J.E., Weil, P.: Profinite semigroups, Mal’cev products and identities. J. Algebra 182, 604–626 (1996)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Reiterman, J.: The Birkhoff theorem for finite algebras. Algebra Universalis 14, 1–10 (1982)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Rhodes, J., Steinberg, B.: The \(q\)-Theory of Finite Semigroups. Springer Monographs in Mathematics. Springer, New York (2009)zbMATHGoogle Scholar
  34. 34.
    Straubing, H.: Aperiodic homomorphisms and the concatenation product of recognizable sets. J. Pure Appl. Algebra 15, 319–327 (1979)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Tilson, B.: Categories as algebra: an essential ingredient in the theory of monoids. J. Pure Appl. Algebra 48, 83–198 (1987)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Trotter, P.G., Weil, P.: The lattice of pseudovarieties of idempotent semigroups and a non-regular analogue. Algebra Universalis 37, 491–526 (1997)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.CMUC, Department of MathematicsUniversity of CoimbraCoimbraPortugal

Personalised recommendations