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

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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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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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