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A quantisation of the calculus of relations. (English) Zbl 0793.06008
Seely, R. A. G. (ed.), Category theory 1991. Proceedings of an international summer category theory meeting, held in Montréal, Québec, Canada, June 23-30, 1991. Providence, RI: American Mathematical Society. CMS Conf. Proc. 13, 345-360 (1992).
Involutive quantales have been recently introduced by the first author in the course of his search for algebraic invariants of \(C^*\)-algebras. In the present paper, this new structure is related to classical algebraic structures appearing in quantum logic. Notably, it is shown that orthocomplemented lattices correspond to certain involutive quantales. This result is closely connected to the representation of orthocomplemented lattices using Foulis semigroups. These involutive quantales generalize the usual quantales of binary relations.
For the entire collection see [Zbl 0771.00047].
Reviewer: J.Rosický (Brno)

06C15 Complemented lattices, orthocomplemented lattices and posets
03G12 Quantum logic
06F05 Ordered semigroups and monoids