Almost Orthogonality and Hausdorff Interval Topologies of de Morgan Lattices and Lattice Effect Algebras

The topologies on ordered structures have been intensively studied by mathematicians and computer scientists. Various types of topologies may be introduced, depending on the nature of the ordered sets considered. Our purpose here is to study the interval topology τ _i , the order topology τ _o and the topology τ _Φ induced by a canonical intrinsic uniformity generated by a certain family of pseudometrics on de Morgan lattices. This uniformity and topology may be regarded as a “two-sided symmetrization” of a similar intrinsic uniformity introduced by Erné and Palko for an order-theoretical construction of certain uniform completions. We prove that on a de Morgan lattice L with a join-dense set $\mathcal{U}$ the interval topology τ _i is Hausdorff and L is compactly generated by the elements of $\mathcal{U}$ if and only if L is $\mathcal{U}$ -almost orthogonal if and only if any element of $\mathcal{U}$ is hypercompact.