Almost Boolean orthomodular posets

Abstract

Let C be the class of concrete (=set-representable) orthomodular partially ordered sets. Let C 0 be the class of Boolean OMP's (Boolean algebras). In-between C 0 and C ( C 0⊂ C ) there are three classes originating in quantum axiomatics — the class C 1 of concrete Jauch-Piron OMP's ( A ϵ C 1 ⇔ if s( A) = s( B) = 1 for a state s on A and A, B ϵ A , then s( C) = 1 for some C ϵ A with C⊂ A∩ B), the class C 2 of ‘compact-like’ OMP's ( A ϵ C 2 ⇔ A is concrete and for every pair A, Bϵ A we have a finite A -covering of A∩ B), and the class C 3 of ‘infimum faithful’ OMP's ( A ϵ C 3 ⇔ if a∧ b = 0 for a, bϵ A then a≤ b′). We study these classes and show that C 0 ⊂ C 1 ⊂ C 2 ⊂ C 3 ⊂ C . We also exhibit examples establishing that at least three of the latter inclusions are proper. Then we prove a representation theorem — every OMP is an epimorphic image of an OMP from C 3. Finally, we comment on the interpretation of the results in quantum axiomatics and formulate open questions.