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On finite simple semigroups

Published online by Cambridge University Press:  20 January 2009

Jorge Almeida
Jorge Almeida
Affiliation:
Inic-Centro de MatemáticaFaculdade de CiênciasUniversidade do Porto4000 PortoPortugal
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Implicit operations (new operations commuting with all old homomorphisms) on pseudovarieties have been shown to play an important role in the study of these classes. They may be used to axiomatize sub-pseudovariaties and to describe recognizable subsets of (relatively) free objects. This paper presents a case study for the pseudovariety CS consisting of all finite simple semigroups. Based on a result of profinite group theory, a structural description of semigroups of implicit operations on finite simple semigroups is used to deduce that CS is join-irreducible.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 34 , Issue 2 , June 1991 , pp. 205 - 215
Copyright
Copyright © Edinburgh Mathematical Society 1991

References

REFERENCES

1.Almeida, J., The algebra of implicit operations, Algebra Universalis 26 (1989), 16–32.CrossRefGoogle Scholar
2.Almeida, J., On pseudovarieties, varieties of languages, filters of congruences, pseudo-identities and related topics, Algebra Universalis 27 (1990), 333–350.CrossRefGoogle Scholar
3.Almeida, J., Residually finite congruences and quasi-regular subsets in uniform algebras, Portugal. Math. 46 (1989), 313–328.Google Scholar
4.Almeida, J., Equations for pseudovarieties, in Pin, J.-E. (Ed.), Formal Properties of Finite Automata and Applications, (Lecture Notes in Comput. Sci. 386, Springer-Verlag, 1989), 148–164.CrossRefGoogle Scholar
5.Almeida, J., Recent developments in the theory of implicit operations (Proceedings of the 1989 Berkeley Workshop on Monoids), to appear.Google Scholar
6.Almeida, J. and Azevedo, A., Implicit operations on certain classes of semigroups, in S. Goberstein and P. Higgins (Eds.), Semigroups and their Applications (Proceedings 1986 Chico Conf., D. Reidel, 1987), 1–11.CrossRefGoogle Scholar
7.Ash, C. J., Pseudovarieties, generalized varieties and similarly described classes, J. Algebra 92 (1985), 104–115.CrossRefGoogle Scholar
8.Banaschewski, B., The Birkhoff Theorem for varieties of finite algebras, Algebra Universalis 17 (1983), 360–368.CrossRefGoogle Scholar
9.Burris, S. and Sankappanavar, H. P., A Course in Universal Algebra (Springer-Verlag, New York, 1981).CrossRefGoogle Scholar
10.Clifford, A. H., The free completely regular semigroup on a set, J. Algebra 59 (1979), 434–451.CrossRefGoogle Scholar
11.Clifford, A. H. and Preston, G. B., The Algebraic Theory of Semigroups, Vol. 1 (Amer. Math. Soc., Providence, 1961).CrossRefGoogle Scholar
12.Eilenberg, S., Automata, Languages and Machines, Vol. B (Academic Press, New York, 1976).Google Scholar
13.Eilenberg, S. and Schotzenberger, M. P., On pseudovarieties, Adv. in Math. 19 (1976), 413–418.CrossRefGoogle Scholar
14.Evans, T., Varieties of semigroups, Semigroup Forum 2 (1971), 1–43.CrossRefGoogle Scholar
15.Fried, M. D., and Jarden, M., Field Arithmetic (Springer-Verlag, Berlin, 1986).CrossRefGoogle Scholar
16.Hall, M. Jr., A topology for free groups and related groups, Ann. of Math. 52 (1950), 127–139.CrossRefGoogle Scholar
17.Lallement, G., Semigroups and Combinatorial Applications (Wiley, New York, 1979).Google Scholar
18.Pin, J.-E., Varieties of Formal Languages (Plenum, London, 1986).CrossRefGoogle Scholar
19.Pin, J.-E., Topologies for the free monoid (LITP report 88–17), (to appear in J. Algebra).Google Scholar
20.Reiterman, J., The Birkhoff theorem for finite algebras, Algebra Universalis 14 (1982), 1–10.CrossRefGoogle Scholar
21.Rhodes, J., New techniques in global semigroup theory, in S. Goberstein and P. Higgins (Eds.), Semigroups and their Applications (Proceedings 1986 Chico Conf., D. Reidel, 1987), 169–181.CrossRefGoogle Scholar