On the set representation of an orthomodular poset
Let P be an orthomodular poset and let B be a Boolean subalgebra of P. A mapping s:P → ⟨0,1⟩ is said to be a centrally additive B-state if it is order preserving, satisfies s(a') = 1 - s(a), is additive on couples that contain a central element, and restricts to a state on B. It is shown that, for any Boolean subalgebra B of P, P has an abundance of two-valued centrally additive B-states. This answers positively a question raised in [13, Open question, p. 13]. As a consequence one obtains a somewhat...