The author considers partially ordered sets with greatest element 1 and least one O: (P,\(\leq,0,1)\) An orthocomplementation is a mapping \(^{\perp}:\) \(p^{\perp \perp}=p\), \(\sup \{p,p^{\perp}\}\simeq 1\), \(p\leq p\to q^{\perp}\leq p^{\perp}\), thus defining an orthoposet, or a quantum logic, respectively. As usual, a subset C of a real is called a cone if \(\{\) \(\lambda\) C:\(\lambda\geq 0\}\subset C\), \(C+C\subset C\). If furthermore \(C\cap -C=0\), there exists a partial order \(\leq_ c\) on \(C:x\leq_ cy\) iff y-x\(\in C\). An order interval is \(\{z:x\leq_ cz\leq_ cy\}\), e is an order unit if \({\mathbb{R}}^{†}\cdot [0,e]={\mathbb{C}}.\)
The main theorem is the following: Let (E,C) be an ordered vector space (E,C) and c be an order unit. Further let M be a set of linear functionals separating points of E fulfilling
(i) \([0,e]_ c\) is \(\sigma (E,M)=compact,\)
(ii) for \(p,q\in Ext[0,e]_ c\) with \(q\nleq p\) there exists a \(\phi\in M\), positive, \(\phi (p)=0\), \(\phi (q)>0\). Then \(Ext[0,e]_ c\) is an orthonormal poset.