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The structure of finite commutative idempotent involutive residuated lattices

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Abstract

We characterize commutative idempotent involutive residuated lattices as disjoint unions of Boolean algebras arranged over a distributive lattice. We use this description to introduce a new construction, called gluing, that allows us to build new members of this variety from other ones. In particular, all finite members can be constructed in this way from Boolean algebras. Finally, we apply our construction to prove that the fusion reduct of any finite member is a distributive semilattice, and to show that this variety is not locally finite.

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References

  1. Anderson, A.R., Belnap, N.D., Jr.: Entailment. Volume I: The Logic of Relevance and Necessity. Princeton University Press, Princeton, London (1975)

    MATH  Google Scholar 

  2. Chajda, I., Halaš, R., Kühr, J.: Semilattice structures, Research and exposition in mathematics, vol. 30. Heldermann, Lemgo (2007)

    MATH  Google Scholar 

  3. Chen, W., Zhao, X.: The structure of idempotent residuated chains. Czechoslov. Math. J. 59(134), 453–479 (2009)

    Article  MathSciNet  Google Scholar 

  4. Chen, W., Zhao, X., Guo, X.: Conical residuated lattice-ordered idempotent monoids. Semigroup Forum 79(2), 244–278 (2009)

    Article  MathSciNet  Google Scholar 

  5. Dunn, J.M.: Algebraic completeness results for \(R\)-mingle and its extensions. J. Symb. Logic 35, 1–13 (1970)

    Article  MathSciNet  Google Scholar 

  6. Frink, O.: Representations of Boolean algebras. Bull. Am. Math. Soc. 47(10), 755–756 (1941)

    Article  MathSciNet  Google Scholar 

  7. Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Elsevier, Amsterdam (2007)

    MATH  Google Scholar 

  8. Gil-Férez, J., Jipsen, P., Metcalfe, G.: Structure theorems for idempotent residuated lattices. Algebra Univ. 81, 1–25 (2020)

    Article  MathSciNet  Google Scholar 

  9. Jenei, S.: Group-representation for even and odd involutive commutative residuated chains (2020). arXiv:1910.01404 (preprint)

  10. Jenei, S.: The Hahn embedding theorem for a class of residuated semigroups. Stud. Log. 108, 1161–1206 (2020)

    Article  MathSciNet  Google Scholar 

  11. McCune, W.: Prover9 and Mace4 (2005–2010). http://www.cs.unm.edu/~mccune/prover9/

  12. Raftery, J.G.: Representable idempotent commutative residuated lattices. Trans. Am. Math. Soc. 359(9), 4405–4427 (2007)

    Article  MathSciNet  Google Scholar 

  13. Stanovský, D.: Commutative idempotent residuated lattices. Czechoslov. Math. J. 57, 191–200 (2007). https://doi.org/10.1007/s10587-007-0055-7

    Article  MathSciNet  MATH  Google Scholar 

  14. Stone, M.H.: Topological representations of distributive lattices and Brouwerian logics. Časopis Pro Pěstování Mat. a Fysiky 67(1), 1–25 (1938). http://eudml.org/doc/27235

  15. Ward, M., Dilworth, R.: Residuated lattices. Trans. Am. Math. Soc. 45, 335–354 (1939)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Some of the computations leading to our results have been obtained with the help of Prover9 and Mace4 [11]. The authors acknowledge the support of funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 689176.

We thank the anonymous referee for many useful remarks and suggestions that improved the manuscript.

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Authors and Affiliations

  1. Faculty of Mathematics, Keck Center of Science and Engineering, Chapman University, 1 University Drive, Orange, CA, 92866, USA

    Peter Jipsen

  2. Mathematical Institute, University of Bern, Sidlerstrasse 5, 3012, Bern, Switzerland

    Olim Tuyt

  3. Dipartimento di Informatica, Università degli Studi di Milano, Via Celoria 18, 20133, Milan, Italy

    Diego Valota

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  1. Peter Jipsen
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  2. Olim Tuyt
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  3. Diego Valota
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Correspondence to Peter Jipsen.

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Communicated by N. Galatos.

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Jipsen, P., Tuyt, O. & Valota, D. The structure of finite commutative idempotent involutive residuated lattices. Algebra Univers. 82, 57 (2021). https://doi.org/10.1007/s00012-021-00751-4

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  • DOI: https://doi.org/10.1007/s00012-021-00751-4

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