Almost Boolean orthomodular posets

AbstractLet C be the class of concrete (=set-representable) orthomodular partially ordered sets. Let C0 be the class of Boolean OMP's (Boolean algebras). In-between C0 and C (C0⊂C) there are three classes originating in quantum axiomatics — the class C1 of concrete Jauch-Piron OMP's (A ϵC1 ⇔ if s(A) = s(B) = 1 for a state s on A and A, B ϵA, then s(C) = 1 for some C ϵ A with C⊂A∩B), the class C2 of ‘compact-like’ OMP's (A ϵC2 ⇔ A is concrete and for every pair A, BϵA we have a finite A-covering of A∩B), and the class C3 of ‘infimum faithful’ OMP's (AϵC3 ⇔ if a∧b = 0 for a, bϵA then a≤b′). We study these classes and show that C0 ⊂ C1 ⊂ C2 ⊂C3 ⊂ 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 fromC3. Finally, we comment on the interpretation of the results in quantum axiomatics and formulate open questions