Ortho and Causal Closure Operations in Ordered Vector Spaces

摘要

On a non-trivial partially ordered real vector space V the orthogonality relation is defined by incomparability and z(V, ^)zeta(V, bot) is a complete lattice of double orthoclosed sets. In an earlier paper we defined an integrally open ordered vector space V and proved orthomodularity of z(V, ^)zeta(V, bot). We shall say that A Í VA subseteq V is an orthogonal set when for all a, b Î Aa, b in A with a ¹ ba neq b, we have a ^ba perp b. We consider two different closure operations A ® A^^A rightarrow A^{perpperp} and A ® D(A)A rightarrow D(A) (ortho and causal closure) and prove: V is integrally open iff A^^ = D(A)A^{perpperp} = D(A) for every orthogonal set A Í VA subseteq V. Hence follows: if V is integrally open, then z(V, ^) = { D(A) Í V : A is an orthogonal set }zeta(V, perp) = { D(A) subseteq V : A {rm,is,an,orthogonal,set} }.