Orthomodular lattices and closure operations in ordered vector spaces
Volume 89 / 2010
Abstract
On a non-trivial partially ordered real vector space $(V, \leq)$ the orthogonality relation is defined by incomparability and $\zeta (V, \perp )$ is a complete lattice of double orthoclosed sets. We say that $ A\subseteq V$ is an orthogonal set when for all $a,b \in A$ with $a \neq b$, we have $a \perp b$. In our earlier papers we defined an integrally open ordered vector space and two closure operations $A \to D(A)$ and $A \to A^{\perp \perp}$. It was proved that $V$ is integrally open iff $D(A)= A^{\perp \perp}$ for every orthogonal set $A \subseteq V$. In this paper we generalize this result. We prove that $V$ is integrally open iff $D(A)=W$ for every $W \in \zeta (V, \perp ) $ and every maximal orthogonal set $A\subseteq W$. Hence it follows that the lattice $\zeta (V, \perp )$ is orthomodular.
