Orthomodular lattices in ordered vector spaces

Abstract

In a partly ordered space the orthogonality relation is defined by incomparability. We define integrally open and integrally semi-open ordered real vector spaces. We prove: if an ordered real vector space is integrally semi-open, then a complete lattice of double orthoclosed sets is orthomodular. An integrally open concept is closely related to an open set in the Euclidean topology in a finite dimensional ordered vector space. We prove: if V is an ordered Euclidean space, then V is integrally open and directed (and is also Archimedean) if and only if its positive cone, without vertex 0, is an open set in the Euclidean topology (and also the family of all order segments {z ∈ V : a < z < b}, a < b, is a base for the Euclidean topology).

Ortho and Causal Closure Operations in Ordered Vector Spaces

“…The equivalences (1) ⇔ (2) ⇔ (3) and implication (1) ⇒ (5) were proved in [3]. (2) is not satisfied.…”

“…Proof. If V is an integrally open space, then by [3] ζ(V, ⊥) is a complete orthomodular lattice. Foulis and Randall [4] proved that ζ(V, ⊥) is a complete orthomodular lattice iff ζ(V, ⊥) = {A ⊥⊥ ⊆ V : A is an orthogonal set}.…”

“…(1) ⇒ (2): Let g be a line, g ∩ P = ∅, 0 / ∈ g and W be the two dimensional subspace generated by g. It was proved in [3] that V is integrally open and directed iff every positive element of V is a strong unite. Hence W is also integrally open and directed, and by [3] W ∩ P \ {0} is an open set in the Euclidean topology of W .…”

“…A set {a + ωb ∈ V : ω ∈ R}, a, b ∈ V, b > 0, is called a P -line (see [3]). Let us denote V P as the family of all P -lines.…”

Orthocomplemented lattices in ordered groups

“…In the paper [2] ( [4]) a partially ordered set Z (an integrally-semi open ordered real vector space V , respectively) was defined and the orthomodularity of the complete lattice E(Z, ⊥) (E(V, ⊥), respectively) was proved. The lattice E(Z, ⊥) is strictly connected to the lattice of the double cones in the Minkowski space-time.…”