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The structure of finite commutative idempotent involutive residuated lattices

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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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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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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Keywords

  • Residuated lattices
  • Substructural logics
  • Boolean algebras
  • Representations
  • Local finiteness

Mathematics Subject Classification

  • 06F05
  • 06F15
  • 03G10
  • 06B15
  • 06E75