Ortho and causal closure operations in ordered vector spaces (Q998765)

Let \(V\) be a partially ordered real vector space. The orthogonality relation \(\perp\) is defined by incomparability. Two closure operators on the subsets of~\(V\) are defined and studied: orthoclosure \(A\mapsto A^{\perp\perp}\) and causal closure \(A\mapsto D(A)= \{a\in V \mid \forall b\in V\), \(b>0 \Rightarrow \exists \omega\in\mathbb{R}\), \(a+\omega b\in A\}\). It is known that the double orthoclosed sets form a complete orthocomplemented poset. This poset is orthomodular if \(V\) is integrally open, i.e., the conjunction of \(b > 0\) and \(a < b\) implies \(\exists n \in N\), \(a + nb > 0\). The inclusion \(D(A)\subseteq A^{\perp\perp}\) always holds. The main result is that \(V\) is integrally open iff the two closure operators coincide on orthogonal sets.