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The structure of generalized BI-algebras and weakening relation algebras

Algebra universalis volume 81, Article number: 35 (2020) Cite this article

Abstract

Generalized bunched implication algebras (GBI-algebras) are defined as residuated lattices with a Heyting implication, and are positioned between Boolean algebras with operators and lattices with operators. We characterize congruences on GBI-algebras by filters that are closed under Gumm–Ursini terms, and for involutive GBI-algebras these terms simplify to a dual version of the congruence term for relation algebras together with two more terms. We prove that representable weakening relation algebras form a variety of cyclic involutive GBI-algebras, denoted by RWkRA, containing the variety of representable relation algebras. We describe a double-division conucleus construction on residuated lattices and on (cyclic involutive) GBI-algebras and show that it generalizes Comer’s double coset construction for relation algebras. Also, we explore how the double-division conucleus construction interacts with other class operators and in particular with variety generation. We focus on the fact that it preserves a special discriminator term, thus yielding interesting discriminator varieties of GBI-algebras, including RWkRA. To illustrate the generality of the variety of weakening relation algebras, we prove that all distributive lattice-ordered pregroups and hence all lattice-ordered groups embed, as residuated lattices, into representable weakening relation algebras on chains. Moreover, every representable weakening relation algebra is embedded in the algebra of all residuated maps on a doubly-algebraic distributive lattice. We give a number of other instructive examples that show how the double-division conucleus illuminates the structure of distributive involutive residuated lattices and GBI-algebras.

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Authors and Affiliations

  1. Department of Mathematics, University of Denver, C.M. Knudson Hall, 2390 S. York St., Denver, CO , 80208, USA

    Nikolaos Galatos

  2. Faculty of Mathematics, Chapman University, One University Drive, Orange, CA , 92866, USA

    Peter Jipsen

Authors
  1. Nikolaos Galatos
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  2. Peter Jipsen
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Correspondence to Peter Jipsen.

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Presented by J.G. Raftery.

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Galatos, N., Jipsen, P. The structure of generalized BI-algebras and weakening relation algebras. Algebra Univers. 81, 35 (2020). https://doi.org/10.1007/s00012-020-00663-9

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  • DOI: https://doi.org/10.1007/s00012-020-00663-9

Keywords

  • Bunched implication algebra
  • Relation algebra
  • Weakening relation
  • Discriminator
  • Substructural logic
  • Residuated lattice

Mathematics Subject Classification

  • 06F05
  • 08B15
  • 03B47
  • 03G10