On semicontinuous lattices and their distributive reflections

In this paper, we are mainly concerned with semicontinuity of complete lattices and their distributive reflections, introduced by Rav in 1989. We prove that for a complete lattice L, the distributive reflection L-d is isomorphic to the lattice of all radicals determined by principal ideals of L in the set-inclusion order, obtaining a method to depict the distributive reflection of a given lattice. It is also proved that if a complete lattice L is semicontinuous and every semiprime element x is an element of L is the largest in d(x), then L-d is continuous whenever the distributive reflector d is Scott continuous. We construct counterexamples to confirm a conjecture and solve two open problems posed by Zhao in 1997.