Topological characterization of certain classes of lattices (Q921025)

Let L be a distributive lattice with 0 and 1, and let Spec L be the set of all proper prime ideals of L. Spec L can be endowed with two topologies, the spectral topology and the D-topology. It is proved that there is a bijection from the set of all \(\sigma\)-ideals of L to the set of all D-open subsets of Spec L. Let Max L and Min L be the sets of maximal ideals and minimal prime ideals of L, respectively. The author proves that the two topologies coincide on Spec L, Max L and Min L, if and only if L is a Boolean, Stonian, and normal lattice, respectively.