The 𝒞(X,D), a characterization of a Stone almost distributive lattice

Abstract

We prove that for any Boolean space [Formula: see text] and a dense Almost Distributive Lattice (ADL) [Formula: see text] with a maximal element, the set [Formula: see text] of all continuous functions of [Formula: see text] into the discrete [Formula: see text] is a Stone ADL. Conversely, it is proved that any Stone ADL is a homomorphic image of [Formula: see text] for a suitable Boolean space [Formula: see text] and a dense ADL [Formula: see text] with a maximal element.