By Isbell duality, each compact regular frame L is isomorphic to the frame of opens of a compact Hausdorff space X. In this note we study the spectrum Spec(L) of prime filters of a compact regular frame L. We prove that X is realized as the minimum of Spec(L) and the Gleason cover of X as the maximum of Spec(L). We also characterize zero-dimensional, extremally disconnected, and scattered compact regular frames by means of Spec(L).
Keywords: Frame, the spectrum of a frame, compact regular frame, compact Hausdorff space, Gleason cover, zero-dimensional frame, extremally disconnected frame, scattered frame
2010 MSC: 06D22; 06D20; 06E15; 54G05; 54G12
Theory and Applications of Categories, Vol. 31, 2016, No. 12, pp 365-383.