Abstract
This article solves a problem proposed by Almeida: the computation of the join of two well-known pseudovarieties of semigroups, namely the pseudovariety of bands and the pseudovariety of locally trivial semigroups. We use a method developed by Almeida, based on the theory of implicit operations.
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References
Albert D., Baldinger R., Rhodes J.,Undecidability of the identity problem for finite semigroups, J. Symbolic Logic,57, 1 (1992), 179–192.
Almeida J., “Semigrupos Finitos e Álgebra Universal”, Publicações do Instituto de Matemática e Estatística da Universidade de São Paulo (1992). “Finite Semigroups and Universal Algebra” (English transl), to appear.
—,Power pseudovarieties of semigroups I, Semigroup Forum,33 (1986), 357–373.
—,Some pseudovariety joins involving the pseudovariety of finite groups, Semigroup Forum,37 (1988), 53–57.
—,The algebra of implicit operations, Algebra Universalis,26 (1989), 16–32.
—,Equations for pseudovarieties, in J.-E. Pin (Ed.) Formal properties of Finite Automata and Applications, Springer, Berlin, Lect. Notes in Computer Science,386 (1989), 148–164.
—,On pseudovarieties, varieties of languages, filters of congruences, pseudoidentities and related topics, Algebra Universalis27 (1990), 333–350.
—,On the membership problem for pseudovarieties of commutative semigroups, Semigroup Forum,42 (1991), 47–51.
Almeida J., Azevedo A.The join of the pseudovarieties of R-trivial and L-trivial monoids, J. Pure and Applied Algebra,60 (1989), 129–137.
—,On regular implicit operations, Portugaliæ Matemática,50, 1 (1993), 35–61.
Almeida J., Weil P.,Reduced factorisations in free profinite groups and join decomposition of pseudovarieties, Int. J. Algebra and Computation.
Almeida J., Weil P.,Relatively free profinite monoids: an introduction and examples, in J. B. Fountain and V. A. R. Gould (Eds.), Semigroups, Formal Languages and Groups, to appear.
Azevedo A.,The join of the pseudovariety J with permutative pseudovarieties, In J. Almeida et al (Eds.), Lattices, Semigroups and Universal Algebra, Plenum, London (1990).
Azevedo A., “Operações Implícitas sobre Pseudovariedades de Semigrupos. Aplicações”, Universidade do Porto, Doctoral dissertation (1989).
—,Operations preserving homomorphisms on the class of finite semigroups DS, Universidade do Porto, Actas 2o Encontro de Algebristas Portugueses, Porto (1987), 33–43.
Biryukov A.P.,Varieties of idempotent semigroups, Algebra Logika,9 (1970), 255–273.
Bourbaki, N., “Éléments de mathématiques”, Topologie généraleI–IV, Hermann, Paris (1971).
Burris S., Sankappanavar H.P., “A Course in Universal Algebra”, Springer, Berlin (1981).
Eilenberg S., “Automata, Languages and Machines”, Academic Press, New York, (A: 1974,B: 1976).
Fennemore C.,All varieties of bands, Semigroup Forum,1 (1970), 172–179.
—,All varieties of bands, Math. Nachr.,48 (1971), 237–262.
Gerhard J. A.,The lattice of equational classes of idempotents semigroups, J. Algebra,15 (1970), 195–224.
Howie J. M., “An Introduction to Semigroup Theory”, Academic Press, London (1976).
Kharlampovich O. G., Sapir M., “Algorithmic Problems in Varieties”, to appear.
Lothaire M., “Combinatorics on Words, Encyclopedia of Mathematics”, Addison Wesley, Reading, MA, 17 (1983).
Pin J.-E., “Variétés de langages formels”, Masson, Paris (1984). “Varieties of Formal Languages” (English transl), Plenum, London (1986).
Polák L.,On varieties of completely regular semigroups I, Semigroup Forum,32 (1985), 97–123.
—,On varieties of completely regular semigroups II, Semigroup Forum,36 (1987), 73–883.
—,On varieties of completely regular semigroups III, Semigroup Forum,37 (1988), 1–30.
Reiterman, J.,The Birkhoff theorem for finite algebras, Algebra Universalis,14 (1982), 1–10.
Rhodes J.,New techniques in global semigroup theory in S. Goberstein and P. Higgins (Eds.),Proc. Chico Conf., Semigroups and their applications, D. Reidel (1987), 25–35.
Trotter P. G., Volkov M. V., The pseudovariety join of\(\mathcal{J}\)-trivial semigroups with groups, to appear.
Volkov M. V.,On a class of semigroup pseudovarieties without finite pseudoidentity basis, Int. J. Algebra and Computation, to appear.
Weil P.,Implicit operations on pseudovarieties: an introduction, in J Rhodes (Ed.), Semigroups and Monoids and Applications, World Scientific, Singapore (1991).
Zeitoun M.,A simple example of a non-finitely based semigroup variety, to appear.
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Communicated by J. E. Pin
This work was partly supported by PRC Mathématiques et Informatique and by ESPRITBRA WG 6317 ASMICS-2
An erratum to this article is available at http://dx.doi.org/10.1007/BF02574101.
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Zeitoun, M. The join of the pseudovarieties of idempotent semigroups and locally trivial semigroups. Semigroup Forum 50, 367–381 (1995). https://doi.org/10.1007/BF02573532
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DOI: https://doi.org/10.1007/BF02573532