Strongly Semicontinuous Lattices

In this paper, we introduce and study strongly semicontinuous lattices, a new class of complete lattices lying between the class of semicontinuous lattices and that of continuous lattices. It is shown that a complete lattice L is strongly semicontinuous iff L is semicontinuous and meet semicontinuous. Some versions of strong interpolation properties for the semiway-below relation in (strongly) semicontinuous lattices are established. Characterization theorems by some distributivity and approximate identities for strongly semicontinuous lattices are given, which reveals that strongly semicontinuous lattices indeed share similar properties with continuous lattices. A subtle counterexample is constructed to show that semicontinuous lattices need not be strongly semicontinuous lattices.