Functorial Polar Functions in Compact Normal Joinfit Frames

KNJ is the category of compact normal joinfit frames and frame homomorphisms. PF is the complete boolean algebra of polars of the frame F. A function X that assigns to each F∈KNJ a subalgebra X(F) of PF that contains the complemented elements of F is a polar function. A polar function X is invariant (resp., functorial) if whenever ϕ:F⟶H∈KNJ is P-essential (resp., skeletal) and p∈X(F), then ϕ(p) ^⊥⊥∈X(H). ϕ:F⟶H∈KNJ is X-splitting if ϕ is P-essential and whenever p∈X(F), then ϕ(p) ^⊥⊥ is complemented in H. F∈KNJ is X-projectable means that every p∈X(F) is complemented. For a polar function X and F∈KNJ, we construct the least X-splitting frame of F. Moreover, we prove that if X is a functorial polar function, then the class of X-projectable frames is a P-essential monoreflective subcategory of KNJS, the category of KNJ-objects and skeletal maps (the case X=P is the result from Martínez and Zenk, which states that the class of strongly projectable KNJ-objects is a reflective subcategory of KNJS).