Inversions on complete lattices and Girard quantales. (English) Zbl 0927.06003

The paper focuses on polarities and quantales. Inversions on complete lattices are introduced and some connections with symmetric relations, quantales and polarities are examined. Girard quantales are described and some algebraic properties of Girard quantales from the point of view of inversions are investigated. A generalization of Girard’s linear logic which takes an advantage of quantales with inversions is constructed. Finally, an application of quantales with inversions in linear logic is given.


06B23 Complete lattices, completions
06F07 Quantales
03G25 Other algebras related to logic
03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
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