ORDER-TOPOLOGICAL COMPLETE ORTHOMODULAR LATTICES

A lattice is order-topological iff its order convergence is topological and makes the lattice operations continuous. We show that the following properties are equivalent for any complete orthomodular lattice L: (i) L is order-topological, (ii) L is continuous (in the sense of Scott), (iii) L is algebraic, (iv) L is compactly atomistic, (v) L is a totally order-disconnected topological lattice in the order topology. A special class of complete order-topological orthomodular lattices, namely the compact topological orthomodular lattices, are characterized by various algebraic conditions, for example, by the existence of a join-dense subset of so-called hypercompact elements.