Remarks on the Stone-Čech and Alexandroff compactifications of locales.

Generalizing a classical result of Smirnov for topological spaces, B. Banaschewski proved that there is an order isomorphism between compactifications of a frame (locale) and the strong inclusions on that frame. Considering, more generally, strong inclusions on distributive pseudocomplemented lattices allows new aspects of locale (and space) compactifications to emerge. Given a subpseudocomplemented lattice P of a frame L and a strong inclusion on P, the 'round ideal completion' of P is a compact regular frame enjoying a unique factorization property for certain continuous mappings from L into compact regular locales. Every compactification can be obtained in this way, from a subpseudocomplemented lattice P and a strong inclusion on P, both inductively defined. It follows that every compactification satisfies (is characterized by) a certain universal property. This property reduces to the usual one of Stone-Čech compactification when the chosen subpseudocomplemented lattice P of L and strong inclusion on P are sufficiently large. One need not take the whole frame (Banaschewski-Mulvey's construction of Stone-Čech compactification). An inductively constructed subpseudocomplemented lattice K of L suffices. Remarkably, it suffices for all compactifications: every compactification of L can be obtained as the frame of round ideals over K for an inductively defined strong inclusion on K. Alexandroff compactification is also re-obtained in this vein, as the completion of the least inductively defined subpseudocomplemented lattice extending a base of L, endowed with the least (inductively defined) strong inclusion containing the way-below relation.