When is the frame of nuclei spatial: A new approach

Abstract

For a frame L, let XL be the Esakia space of L. We identify a special subset YL of XL consisting of nuclear points of XL, and prove the following results: • L is spatial iff YL is dense in XL. • If L is spatial, then N(L) is spatial iff YL is weakly scattered. • If L is spatial, then N(L) is boolean iff YL is scattered. As a consequence, we derive the well-known results of Beazer and Macnab [1], Simmons [22], Niefield and Rosenthal [13], and Isbell [10].