Cozero bases of frames

A frame L which is generated by a regular σ-frame (= cozero basis of L) is completely regular. Its cozero part Coz L is then the largest cozero basis of L, and we characterize here those L for which it is the only such. Further, we give a similar characterization for the finitary analogue of this situation where regular σ-frames are replaced by Boolean algebras. In addition, we consider the compactifications of a frame L provided by its cozero bases and show that all compactifications are of this kind iff L is pseudocompact. Finally, as an aside, we characterize the completely regular frames with unique compactification and their zero-dimensional counterparts.