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Three roads to complete lattices: orders, compatibility, polarity. (English) Zbl 1486.06004

In this paper, the author clarifies the relations between three ways of constructing complete lattices that appear in three different areas: (1) using ordered structures, as in set-theoretic forcing, or doubly ordered structures, as in a recent semantics for intuitionistic logic; (2) using compatibility relations, as in semantics for quantum logic based on ortholattices; (3) using Birkhoff’s polarities, as in formal concept analysis.

MSC:

06B23 Complete lattices, completions
06C15 Complemented lattices, orthocomplemented lattices and posets
06D20 Heyting algebras (lattice-theoretic aspects)
03G05 Logical aspects of Boolean algebras
03G10 Logical aspects of lattices and related structures
03G12 Quantum logic
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References:

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