Epicompletion in Frames with Skeletal Maps, I: Compact Regular Frames

Abstract

A frame homomorphism h : A ⟶ B is skeletal if x⊥⊥ = 1 in A implies that h(x)⊥⊥ = 1 in B. It is shown that, in \(\mathfrak{KRegS}\), the category of compact regular frames with skeletal maps, the subcategory \(\mathfrak{SPRegS}\), consisting of the frames in which every polar is complemented, coincides with the epicomplete objects in \(\mathfrak{KRegS}\). Further, \(\mathfrak{SPRegS}\) is the least epireflective subcategory, and, indeed, the target of the monoreflection which assigns to a compact regular frame A, the ideal frame e A of \(\mathcal{P} A\), the boolean algebra of polars of A.