Functorial Polar Functions in Compact Normal Joinfit Frames

Abstract

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