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 . It was proved that V is integrally open iff 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc89-0-7,
author = {Jan Florek},
title = {Orthomodular lattices and closure operations in ordered vector spaces},
journal = {Banach Center Publications},
volume = {89},
year = {2010},
pages = {129-133},
zbl = {1223.06008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc89-0-7}
}
Jan Florek. Orthomodular lattices and closure operations in ordered vector spaces. Banach Center Publications, Tome 89 (2010) pp. 129-133. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc89-0-7/