Pointfree Functorial Polar Functions

Abstract

C denotes the category of compact regular frames with frame homomorphisms. A function \(\mathcal {X}\), which assigns to each C-object F a subalgebra of \(\mathcal {P}(F)\) that contains the complemented elements of F is said to be a polar function. An essential extension H of F is a \(\mathcal {X}\)-splitting frame of F if whenever \(p \in \mathcal {X}(F)\), then the polar generated by p in H is complemented. For F∈ C we examine the least \(\mathcal {X}\)-splitting extension and prove that every invariant polar function generates a C-hull class of frames. In addition, we define the concept of a functorial polar function and prove that each functorial polar function generates an epireflective subcategory of the category compact regular frames with skeletal maps.