On the set representation of an orthomodular poset (Q2759166)

Let \(P\) be an orthomodular poset and let \(B\) be a Boolean subalgebra of \(P\). Let us call a mapping \(s\colon P\to[0,1]\) a centrally additive \(B\)-state if \(s\) is order preserving, satisfies \(s(a')=1-s(a)\) for each \(a\in P\), is additive on couples that contain a central element, and restricts to a Boolean state on \(B\). The authors show that for any Boolean subalgebra of \(P\), \(P\) always possesses an abundance of two-valued centrally additive \(B\)-states. This solves a problem formulated by \textit{J. Tkadlec} [``Partially additive states on orthomodular posets'', Colloq. Math. 62, 7-14 (1991; Zbl 0784.03037)] and allows one to find a generalized set-representation theorem for orthomodular posets and orthomodular lattices. Further generalization is hardly possible as the authors demonstrate by an example.