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: 1. is order-topological, is continuous (in the sense of Scott), L is algebraic, is compactly atomistic, is a totally order-disconnected topological lattice in the order topology. ial 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.