Hull classes in compact regular frames

Abstract

\(\mathfrak {KReg}\) is the category of compact regular frames and frame homomorphisms. A class of \(\mathfrak {KReg}\) frames \(\textbf{H}\) is a hull class provided that: (i) \(\textbf{H}\) is closed under isomorphic copies; (ii) for every \(F \in \mathfrak {KReg}\) there exist an \(hF \in \textbf{H}\) and a morphism \(h_F\) such that \(F \overset{h_F}{\le }\ hF\) is essential; (iii) if \(F \overset{\phi }{\le }\ H\) is essential and \(H \in \textbf{H}\), then there exists \(h\phi : hF \longrightarrow H\) for which \(\phi = h\phi \cdot h_F\). This work provides techniques for identifying and generating hull classes in \(\mathfrak {KReg}\). Moreover, for a compact regular frame F, we introduce and investigate various properties of projectability and disconnectivity of F and prove that for each property, P, the class of \(\mathfrak {KReg}\)-objects that satisfy P is a hull class in \(\mathfrak {KReg}\). In addition, we provide examples of \(\mathfrak {KReg}\) hull classes that are not characterized by some form of projectability/disconnectivity and examples of classes of \(\mathfrak {KReg}\)-objects that are not hull classes.