The Freudenthal and other compactifications of continuous frames

Abstract

The N-star compactifications of frames are the frame-theoretic counterpart of the N-point compactifications of locally compact Hausdorff spaces. A \(\pi \)-compactification of a frame L is a compactification constructed using a special type of a basis called a \(\pi \)-compact basis; the Freudenthal compactification is the largest \(\pi \)-compactification of a rim-compact frame. As one of the main results, we show that the Freudenthal compactification of a regular continuous frame is the least upper bound for the set of all N-star compactifications. A compactification whose right adjoint preserves disjoint binary joins is called perfect. We establish a class of frames for which N-star compactifications are always perfect. For the class of zero-dimensional frames, we construct a compactification which is isomorphic to the Banaschewski compactification and the Freudenthal compactification; in some special case, this compactification is isomorphic to the Stone–Čech compactification.