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The Birkhoff theorem for finite algebras

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Abstract

A finite analogue of the Birkhoff variety theorem is proved: a non-void class of finite algebras of a finite type τ is closed under the formation of finite products, subalgebras and homomorphic images if and only if it is definable by equations for implicit operations, that is, roughly speaking, operations which are not necessarily induced by τ-terms but which are compatible with all homomorphisms. It is well-known that explicit operations (those induced by τ-terms) do not suffice for such an equational description. Topological aspects of implicit operations are considered. Various examples are given.

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References

  1. J. T. Baldwin andJ. Berman,Varieties and finite closure conditions, Coll. Math.35 (1976), 15–20.

    MATH  MathSciNet  Google Scholar 

  2. B. Banaschewski,On equationally compact extensions of algebras, Alg. Univ.4 (1974), 20–35.

    MATH  MathSciNet  Google Scholar 

  3. B. Banaschewski,Profinite universal algebras, in: Proceedings of the Third Prague Topological Symposium, Academia Prague 1971, 51–62.

  4. G. Birkhoff,On the structure of abstract algebras, Proc. Cambridge Phil. Soc.31 (1935), 433–454.

    Article  MATH  Google Scholar 

  5. S. Bulman-Fleming,Congruence topologies on universal algebras, Math. Z.119 (1971), 287–289.

    Article  MathSciNet  Google Scholar 

  6. P. M. Cohn,Universal algebra, Harper and Row, New York, Evanston and London 1965.

    MATH  Google Scholar 

  7. S. Eilenberg, Automata, languages and machines, Volume B, Academic Press, New York, San Francisco, London 1976.

    Google Scholar 

  8. S. Eilenberg andM. P. Schützenberger,On pseudovarieties, Advances in Math.19 (1976), 413–418.

    Article  MATH  MathSciNet  Google Scholar 

  9. T. Evans,Approximating algebras by finite algebras, Seminaire Mathématique Superieures, L'Université de Montréal 1971.

  10. P. Goralčík andV. Koubek,Pseudovarieties and pseudotheories, to appear in Proceedings of Colloquium on Finite Algebra and Multiple Valued Logic, Szeged 1979.

  11. G. Grätzer,Universal algebra, D. van Nostrand, Princeton 1968.

    MATH  Google Scholar 

  12. M. Hull,A topology for free groups and related groups, Ann. of Math. (2)52 (1950), 127–139.

    Article  MathSciNet  Google Scholar 

  13. J. F. Kennison andD. Gildenhuys,Equational completions, model induced triples and pro-objects, J. Pure Appl. Algebra1 (1971), 317–346.

    Article  MATH  MathSciNet  Google Scholar 

  14. F. W. Lawvere, Functorial semantics of algebraic theories, dissertation, Columbia University 1963.

  15. F. W. Lawvere, Some algebraic problems in the context of functorial semantics of algebraic theories, in: Lecture Notes in Mathematics 61, Springer-Verlag 1968, 41–61.

  16. V. Müller: Algebras and R-algebras, thesis, Charles University Prague 1973.

  17. J. Mycielski andW. Taylor,A compactification of the algebra of terms, Alg. Univ.6 (1976), 161–163.

    MathSciNet  Google Scholar 

  18. J. Rosický andL. Polák,Definability on finite algebras, to appear in Proceedings of Colloquium on Finite Algebra and Multiple Valued Logic, Szeged 1979.

  19. W. Taylor,Some constructions of compact algebras, Ann. Math. Logic3 (1971), 395–435.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Technical University of Prague, Prague, Czechoslovakia

    Jan Reiterman

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  1. Jan Reiterman
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Reiterman, J. The Birkhoff theorem for finite algebras. Algebra Universalis 14, 1–10 (1982). https://doi.org/10.1007/BF02483902

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  • DOI: https://doi.org/10.1007/BF02483902

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