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Relation Algebras, Idempotent Semirings and Generalized Bunched Implication Algebras

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Relational and Algebraic Methods in Computer Science (RAMICS 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10226))

Abstract

This paper investigates connections between algebraic structures that are common in theoretical computer science and algebraic logic. Idempotent semirings are the basis of Kleene algebras, relation algebras, residuated lattices and bunched implication algebras. Extending a result of Chajda and Länger, we show that involutive residuated lattices are determined by a pair of dually isomorphic idempotent semirings on the same set, and this result also applies to relation algebras. Generalized bunched implication algebras (GBI-algebras for short) are residuated lattices expanded with a Heyting implication. We construct bounded cyclic involutive GBI-algebras from so-called weakening relations, and prove that the class of weakening relation algebras is not finitely axiomatizable. These algebras play a role similar to representable relation algebras, and we identify a finitely-based variety of cyclic involutive GBI-algebras that includes all weakening relation algebras. We also show that algebras of down-closed sets of partially-ordered groupoids are bounded cyclic involutive GBI-algebras.

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Authors and Affiliations

  1. Chapman University, Orange, CA, 92866, USA

    Peter Jipsen

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  1. Peter Jipsen
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Correspondence to Peter Jipsen .

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Editors and Affiliations

  1. Data61, CSIRO, Sydney, New South Wales, Australia

    Peter Höfner

  2. CNRS, Lyon, France

    Damien Pous

  3. University of Sheffield, Sheffield, United Kingdom

    Georg Struth

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Jipsen, P. (2017). Relation Algebras, Idempotent Semirings and Generalized Bunched Implication Algebras. In: Höfner, P., Pous, D., Struth, G. (eds) Relational and Algebraic Methods in Computer Science. RAMICS 2017. Lecture Notes in Computer Science(), vol 10226. Springer, Cham. https://doi.org/10.1007/978-3-319-57418-9_9

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  • DOI: https://doi.org/10.1007/978-3-319-57418-9_9

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-57417-2

  • Online ISBN: 978-3-319-57418-9

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