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 \to \langle 0, 1 \rangle $ 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 better set representation of orthomodular posets and a better extension theorem than in [2, 12, 13]. Further improvement in the Boolean vein is hardly possible as the concluding example shows.