Orthomodular lattices and closure operations in ordered vector spaces

Jan Florek

Banach Center Publications (2010)

  • Volume: 89, Issue: 1, page 129-133
  • ISSN: 0137-6934

Abstract

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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.

How to cite

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Jan Florek. "Orthomodular lattices and closure operations in ordered vector spaces." Banach Center Publications 89.1 (2010): 129-133. <http://eudml.org/doc/281698>.

@article{JanFlorek2010,
abstract = {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.},
author = {Jan Florek},
journal = {Banach Center Publications},
keywords = {ordered vector space; orthogonality space; orthomodular lattice; closure operator; strong unit; order topology},
language = {eng},
number = {1},
pages = {129-133},
title = {Orthomodular lattices and closure operations in ordered vector spaces},
url = {http://eudml.org/doc/281698},
volume = {89},
year = {2010},
}

TY - JOUR
AU - Jan Florek
TI - Orthomodular lattices and closure operations in ordered vector spaces
JO - Banach Center Publications
PY - 2010
VL - 89
IS - 1
SP - 129
EP - 133
AB - 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.
LA - eng
KW - ordered vector space; orthogonality space; orthomodular lattice; closure operator; strong unit; order topology
UR - http://eudml.org/doc/281698
ER -

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