The author has recently
introduced the generalized interval topology on a partially ordered set as an
alternative to the standard interval topology. In this paper, the structure of
generalized segments in lattices is investigated, and sufficient conditions are given for
the generalized interval topology on a distributive lattice to be a lattice topology;
adding another condition ensures that the topology is Hausdorff. Similar results are
obtained for a slight modification of the generalized interval topology, the generalized
star-interval topology, and examples are constructed which illustrate less restrictive
situations.
