Ortho and Causal Closure Operations in Ordered Vector Spaces期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Ortho and Causal Closure Operations in Ordered Vector Spaces

Authors:	Jan Florek

Affiliation:	(1) Institute of Mathematics, University of Economics, ul. Komandorska 118/120, 53-345 Wrocław, Poland

Abstract:	On a non-trivial partially ordered real vector space V the orthogonality relation is defined by incomparability and $\zeta(V, \bot)$ 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 $\zeta(V, \bot)$ . We shall say that $A \subseteq V$ is an orthogonal set when for all $a, b \in A$ with $a \neq b$ , we have $a \perp b$ . We consider two different closure operations $A \rightarrow A^{\perp\perp}$ and $A \rightarrow D(A)$ (ortho and causal closure) and prove: V is integrally open iff $A^{\perp\perp} = D(A)$ for every orthogonal set $A \subseteq V$ . Hence follows: if V is integrally open, then $\zeta(V, \perp) = \{ D(A) \subseteq V : A {\rm\,is\,an\,orthogonal\,set} \}$ . Received July 6, 2007; accepted in final form July 31, 2007.

Keywords:	and phrases: ordered vector space orthogonality space orthomodular lattice
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