Abstract
Grishin algebras are a generalisation of Boolean algebras that provide algebraic models for classical bilinear logic with two mutually cancelling negation connectives. We show how to build complete Grishin algebras as algebras of certain subsets (“propositions”) of cover systems that use an orthogonality relation to interpret the negations.
The variety of Grishin algebras is shown to be closed under MacNeille completion, and this is applied to embed an arbitrary Grishin algebra into the algebra of all propositions of some cover system, by a map that preserves all existing joins and meets.
This representation is then used to give a cover system semantics for a version of classical bilinear logic that has first-order quantifiers and infinitary conjunctions and disjunctions.
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Dedicated to Ryszard Wójcicki on his 80th birthday
Special issue in honor of Ryszard Wójcicki on the occasion of his 80th birthday Edited by J. Czelakowski, W. Dziobiak, and J. Malinowski
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Goldblatt, R. Grishin Algebras and Cover Systems for Classical Bilinear Logic. Stud Logica 99, 203 (2011). https://doi.org/10.1007/s11225-011-9360-0
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DOI: https://doi.org/10.1007/s11225-011-9360-0