Lattices and quantum logics with separated intervals, atomicity
A bounded lattice is said to have separated intervals if it admits, for any two disjoint intervals \([a,b]\) and \([c,d]\), a finite covering that consists of closed intervals each of which is disjoint with at least one of the intervals \([a,b]\) and \([c,d]\). The author investigates connections between atomicity of a bounded lattice \(L\), the interval topology on \(L\) and on the MacNeille completion of \(L\), and the condition that \(L\) has separated intervals. For example, a complete lattice \(L\) satisfies the condition iff the interval topology on \(L\) is Hausdorff. Some other results: An atomistic ortholattice has separated intervals iff the interval topology on its MacNeille completion is Hausdorff. A complete orthomodular lattice having separated intervals is atomic. A Boolean lattice has separated intervals iff it is atomic.
scientific article; zbMATH DE number 1196982
scientific article