Orthomodular lattices in ordered vector spaces (Q868727)

Let \(V\) be a partially ordered real vector space. The orthogonality relation is defined by incomparability. The space \(V\) is called integrally semi-open if \(a>0\) and \(a>b\) implies that \(\exists n\in\mathbb{N}\), \(na+b>0\) or \(\forall n\in\mathbb{N}\), \(na+b\leq 0\). (An equivalent characterization in terms of positive cones is also given.) The main result is that, for a convex set \(Z\subseteq V\), the complete lattice of its double orthoclosed subsets is orthomodular.