Boolean orthoposets and two-valued states on them (Q1308491)

Let \(P\) be an orthoposet. If \(a\vee b=0\) \((a,b\in P)\) implies \(a\perp b\), we call \(P\) Boolean. A state (bounded non-negative measure) \(s\) on \(P\) is called Jauch-Piron if for every pair \(a,b\in P\) with \(s(a)=1=s(b)\) there is a \(c\in P\) with \(s(c)= 1\) such that \(c\leq a,b\). The author proves that: (i) Every orthoposet with a full set of two-valued Jauch-Piron states is Boolean, and (ii) Every atomic Boolean orthoposet has a full set of two-valued Jauch-Piron states. This is not true for orthomodular posets.