Šmarda, Bohumil Inversions on complete lattices and Girard quantales. (English) Zbl 0927.06003 Acta Math. Inform. Univ. Ostrav. 5, No. 1, 53-59 (1997). 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. Reviewer: R.Smolíková (Ostrava) MSC: 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) Keywords:polarity; quantale; inversion; complete lattices; linear logic PDF BibTeX XML Cite \textit{B. Šmarda}, Acta Math. Inform. Univ. Ostrav. 5, No. 1, 53--59 (1997; Zbl 0927.06003) Full Text: EuDML OpenURL References: [1] Birkhoff G.: Lattice Theory. (Third Edition), Amer. Math. Soc. Providence. R. J., 1973. · Zbl 0126.03801 [2] Girard J. Y.: Linear Logic. Theor.Comp.Sci., 50 (1987), 1-102. · Zbl 0647.03016 [3] Rosenthal K. J.: Quantales and their applications. Longman Sci and Techn. New York, 1990. · Zbl 0703.06007 [4] Šmarda B.: Polarities on algebraical structures. Acta Math. Univ. Com., 58(1991), 181-187. · Zbl 0743.20042 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.