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Canonical completions of lattices and ortholattices. (English) Zbl 0939.06004

MacNeille completions of posets have a bad behaviour with respect to the orthomodular law. It is still an open question whether every orthomodular lattice can be embedded into a complete orthomodular lattice. The author shows that every bounded lattice \(L\) has a completion which is determined up to isomorphism by a given system of six axioms. Unfortunately, this so-called canonical completion of \(L\) need not be orthomodular (modular), even if \(L\) is orthomodular (modular). On the other hand a canonical completion of an ortholattice is a complete ortholattice and if \(L\) is distributive then the canonical completion is also distributive. Moreover, it is shown that any hope of finding and orthomodular completion must necessarily abandon regularity (i.e., the preservation of all existing joins and meets).

MSC:

06B23 Complete lattices, completions
06C15 Complemented lattices, orthocomplemented lattices and posets
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06E25 Boolean algebras with additional operations (diagonalizable algebras, etc.)
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