On some parallelism between complete regularity and zero-dimensionality

Abstract

We explore some parallelism between the categories CRFrm and 0DFrm of completely regular frames and zero-dimensional frames, respectively, with a view to establishing zero-dimensional analogues of C*-quotients. A lattice homomorphism between the cozero parts of two completely regular frames can be lifted to a frame homomorphism between the Stone-Čech compactifications of the frames involved [13]. Here we lift a lattice homomorphism ψ: BL → BM between the Boolean parts of two zero-dimensional frames to a frame homomorphism between their universal zero-dimensional compactifications, and then study some properties of the lift.