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Structure theorems for idempotent residuated lattices

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In this paper we study structural properties of residuated lattices that are idempotent as monoids. We provide descriptions of the totally ordered members of this class and obtain counting theorems for the number of finite algebras in various subclasses. We also establish the finite embeddability property for certain varieties generated by classes of residuated lattices that are conservative in the sense that monoid multiplication always yields one of its arguments. We then make use of a more symmetric version of Raftery’s characterization theorem for totally ordered commutative idempotent residuated lattices to prove that the variety generated by this class has the amalgamation property. Finally, we address an open problem in the literature by giving an example of a noncommutative variety of idempotent residuated lattices that has the amalgamation property.

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Authors and Affiliations

  1. Mathematical Institute, University of Bern, Alpeneggstrasse 22, 3012, Bern, Switzerland

    José Gil-Férez

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

    Peter Jipsen

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

    George Metcalfe

Authors
  1. José Gil-Férez
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  2. Peter Jipsen
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  3. George Metcalfe
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Correspondence to George Metcalfe.

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The research of the second and third authors was supported by the Swiss National Science Foundation (SNF) grant 200021_165850.

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Gil-Férez, J., Jipsen, P. & Metcalfe, G. Structure theorems for idempotent residuated lattices. Algebra Univers. 81, 28 (2020). https://doi.org/10.1007/s00012-020-00659-5

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  • DOI: https://doi.org/10.1007/s00012-020-00659-5

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