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A relatively finite-to-finite universal but not Q-universal quasivariety

Algebra universalis volume 83, Article number: 26 (2022) Cite this article

Abstract

It was proved by the authors that the quasivariety of quasi-Stone algebras \(\mathbf {Q}_{\mathbf {1,2}}\) is finite-to-finite universal relative to the quasivariety \(\mathbf {Q}_{\mathbf {2,1}}\) contained in \(\mathbf {Q}_{\mathbf {1,2}}\). In this paper, we prove that \(\mathbf {Q}_{\mathbf {1,2}}\) is not Q-universal. This provides a positive answer to the following long standing open question: Is there a quasivariety that is relatively finite-to-finite universal but is not Q-universal?

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References

  1. Adams, M.E., Adaricheva, K.V., Dziobiak, W., Kravchenko, A.V.: Some open questions related to the problem of Birkhoff and Maltsev. Stud. Logica. 78, 357–378 (2004)

    Article  Google Scholar 

  2. Adams, M.E., Dziobiak, W.: Finite-to-finite universal quasivarieties are \(Q\)-universal. Algebra Univers. 46, 253–283 (2001)

    MathSciNet  Article  Google Scholar 

  3. Adams, M.E., Dziobiak, W.: Remarks about the \(Q\)-lattice of the variety of lattices. Algebra Univers. 82 (2021)

  4. Adams, M.E., Dziobiak, W., Kravchenko, A.V., Schwidefsky, M.V: Remarks about complete lattice homomorphic images of algebraic lattices (2020)

  5. Adams, M.E., Dziobiak, W., Sankappanavar, H.P.: Universal varieties of quasi-Stone algebras. Algebra Univers. 76, 155–182 (2016)

    MathSciNet  Article  Google Scholar 

  6. Birkhoff, G.: Universal algebra. In: Proceedings of the First Canadian Math. Congress (Montreal, 1945), 310–326. The University of Toronto Press, Toronto (1946)

  7. Fischer, S.K.: Amalgamation in the varieties of quasi-Stone algebras, PhD thesis, University of Bern (2011)

  8. Fraser, G.A., Horn, A.: Congruence relations in direct products. Proc. Am. Math. Soc. 26, 390–394 (1970)

    MathSciNet  Article  Google Scholar 

  9. Freese, R., Ježek, J., Nation, J.B.: Free Lattices. Mathematical Surveys and Monographs vol. 42. American Mathematical Society, Providence, RI (1995)

  10. Gaitán, G.: Priestley duality for quasi-Stone algebras. Stud. Log. 64, 83–92 (2000)

    MathSciNet  Article  Google Scholar 

  11. Gorbunov, V.A.: Algebraic Theory of Quasivarieties. Plenum Publishing Co., New York (1998)

    MATH  Google Scholar 

  12. Grätzer, G., Kelly, D.: Subdirectly irreducible members of products of lattice varieties. Proc. Am. Math. Soc. 102, 483–489 (1988)

    MathSciNet  Article  Google Scholar 

  13. Z. Hedrlín, Z., Pultr, A.: On full embeddings of categories of algebras. IL. J. Math. 10, 392–406 (1966)

  14. Hyndman, J., Nation, J.B.: The Lattice of Subquasivarieties of a Locally Finite Quasivariety. Springer, New York (2018)

  15. Koubek, V., Sichler, J.: On relative universality and Q-universality. Stud. Logic 78, 279–291 (2004)

    MathSciNet  Article  Google Scholar 

  16. Koubek, V., Sichler, J.: Almost \(\mathit{ff}\)-universal and Q-universal varieties of modular 0-lattices. Colloq. Math. 101, 161–182 (2004)

    MathSciNet  Article  Google Scholar 

  17. Koubek, V., Sichler, J.: On synchronized relatively full embeddings and Q-universality. Cah. Topol. Géom. Différ. Catég. 49, 289–306 (2008)

    MathSciNet  MATH  Google Scholar 

  18. Koubek, V., Sichler, J.: Almost \(\mathit{ff}\)-universality implies \(Q\)-universality. Appl. Categ. Struct. 17, 419–434 (2009)

    MathSciNet  Article  Google Scholar 

  19. Koubek, V., Sichler, J.: On relative universality and Q-universality of finitely generated varieties of Heyting algebras. Sci. Math. Jpn. 74, 63–115 (2011)

    MathSciNet  MATH  Google Scholar 

  20. Kravchenko, A.V., Nurakunov, A.M., Schwidefsky, M.V.: On the structure of quasivarieties. I. Independent axiomatizability. Algebra Logic 57, 445–462 (2019)

  21. Kravchenko, A.V., Nurakunov, A.M., Schwidefsky, M.V.: On the structure of quasivarieties. II. Undecidable problems. Algebra Logic 58, 123–136 (2019)

  22. Libkin, L.: n-Distributivity, dimension and Carathéodory’s theorem. Algebra Univers. 34, 72–95 (1995)

    Article  Google Scholar 

  23. Maltsev, A.I.: Multiplication of classes of algebraic systems. Sibirsk. Mat. Ž. 8, 346–365 (1967). (Russian)

    MathSciNet  Google Scholar 

  24. Maltsev, A.I.: Problems on the Borderline of Algebra and Logic, pp. 217–231. Proc. Inter. Congress of Mathematicians, Moscow (1968)

    Google Scholar 

  25. Priestley, H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bull. Lond. Math. Soc. 2, 186–190 (1970)

    MathSciNet  Article  Google Scholar 

  26. Pultr, P., Trnková, V.: Combinatorial, Algebraic and Topological Representations of Groups. Semigroups and Categories. North-Holland, Amsterdam (1980)

    MATH  Google Scholar 

  27. Sankappanavar, N.H., Sankappanavar, H.P.: Quasi-Stone algebras. Math. Logic Quart. 39, 255–268 (1993)

    MathSciNet  Article  Google Scholar 

  28. Sapir, M.V.: The lattice of quasivarieties of semigroups. Algebra Univers. 21, 172–180 (1985)

    MathSciNet  Article  Google Scholar 

  29. Schwidefsky, M.V.: Existence of independent quasi-equational bases. Algebra Logic 58, 514–537 (2020)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

We thank Bill Sands for his correspondence with us related to the results presented here. We also thank Sara-Kaja Fischer whose thesis [7] helped us to refresh our interest in quasi-Stone algebras.

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Authors and Affiliations

  1. Department of Mathematics, State University of New York, New Paltz, NY, 12561, USA

    M. E. Adams & H. P. Sankappanavar

  2. Department of Mathematics, University of Puerto Rico, Mayagüez, PR, 00681, Puerto Rico

    W. Dziobiak

Authors
  1. M. E. Adams
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  2. W. Dziobiak
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  3. H. P. Sankappanavar
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Corresponding author

Correspondence to M. E. Adams.

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Communicated by Presented by E. W. H. Lee.

To the memory of Jaroslav Ježek.

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The results of this paper were presented by the second author to the audience of the Maltsev Meeting held in August 19–23, 2019, Novosibirsk (Russia)

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Adams, M.E., Dziobiak, W. & Sankappanavar, H.P. A relatively finite-to-finite universal but not Q-universal quasivariety. Algebra Univers. 83, 26 (2022). https://doi.org/10.1007/s00012-022-00782-5

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  • DOI: https://doi.org/10.1007/s00012-022-00782-5

Keywords

  • Quasivariety
  • Q-lattice
  • Relative finite-to-finite universality
  • Q-universality
  • Quasi-stone algebras

Mathematics Subject Classification

  • Primary: 06B05
  • 08C15
  • Secondary: 06B20