Orthomodular lattices and closure operations in ordered vector spaces

• Autorzy: Jan Florek
• Języki publikacji: EN

Abstrakty

EN

On a non-trivial partially ordered real vector space (V,≤) the orthogonality relation is defined by incomparability and ζ(V,⊥) is a complete lattice of double orthoclosed sets. We say that A ⊆ V is an orthogonal set when for all a,b ∈ A with a ≠ b, we have a ⊥ b. In our earlier papers we defined an integrally open ordered vector space and two closure operations A → D(A) and $A → A^{⊥⊥}$. It was proved that V is integrally open iff $D(A) = A^{⊥⊥}$ for every orthogonal set A ⊆ V. In this paper we generalize this result. We prove that V is integrally open iff D(A) = W for every W ∈ ζ(V,⊥) and every maximal orthogonal set A ⊆ W. Hence it follows that the lattice ζ(V,⊥) is orthomodular.

Kategorie tematyczne

• 06F20: Ordered abelian groups, Riesz groups, ordered linear spaces
• 06C15: Complemented lattices, orthocomplemented lattices and posets
• 06B30: Topological lattices, order topologies

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2010
• Tom: 89
• Numer: 1
• Strony: 129-133

Daty

wydano

2010

Twórcy

autor

Jan Florek

• Institute of Mathematics, University of Economics, Komandorska 118/120, 53-345 Wrocław, Poland

Identyfikatory

DOI

10.4064/bc89-0-7