Orthocomplemented lattices in ordered groups

Abstract

On a partially ordered set G the orthogonality relation $$\bot$$ is defined by incomparability and $${\mathfrak E}(G, \bot)$$ is a complete orthocomplemented lattice of double orthoclosed sets. We will prove that the atom space $$\Omega({\mathfrak E})$$ of the lattice $${\mathfrak E}(G, \bot)$$ has the same order structure as G. Thus if G is a partially ordered set (an ordered group, or an ordered vector space), then $$\Omega({\mathfrak E})$$ is a canonically partially ordered set (an ordered quotient group, or an ordered quotient vector space, respectively). We will also prove: if G is an ordered group with a positive cone P, then the lattice $${\mathfrak E}(G, \bot)$$ has the covering property iff $$P-P = -P{\cup}P{\cup}(g+M)$$ , where g is an element of G and M is the intersection of all maximal subgroups contained in $$-P \cup P$$ .