Hull Classes in Compact Regular Frames

𝔎ℜ𝔢𝔤 is the category of compact regular frames and frame homomorphisms. A class of 𝔎ℜ𝔢𝔤 frames H is a hull class provided that: (i) H is closed under isomorphic copies; (ii) for every F ∈𝔎ℜ𝔢𝔤 there exist an hF ∈H and a morphism h_F such that F h_F≤ hF is essential; (iii) if F ϕ≤ H is essential and H ∈H , then there exists hϕ : hF ⟶ H for which ϕ = hϕ· h_F . This work provides techniques for identifying and generating hull classes in 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 𝔎ℜ𝔢𝔤 -objects that satisfy P is a hull class in 𝔎ℜ𝔢𝔤 . In addition, we provide examples of 𝔎ℜ𝔢𝔤 hull classes that are not characterized by some form of projectability/disconnectivity and examples of classes of 𝔎ℜ𝔢𝔤 -objects that are not hull classes.