Morphisms and Pushouts in Compact Normal Joinfit Frames

Abstract

$\mathfrak {KNJ}$ is the category of compact normal joinfit frames and frame homomorphisms and $\mathfrak {KReg}$ is the coreflective subcategory of compact regular frames. This work investigates $\mathfrak {KNJ}$ through its interaction with $\mathfrak {KReg}$ via the coreflection $\rho$. A $\mathfrak {KNJ}$ morphism $\phi : F \longrightarrow M$ is $\mathcal {P}$-essential if $\phi$ is skeletal and the map between the frames of polars, $\mathcal {P}(\phi ): \mathcal {P}F \longrightarrow \mathcal {P}M$ defined by $\mathcal {P}(\phi )(p)=\phi (p)^{\perp \perp }$, is a boolean isomorphism. The $\mathcal {P}$-essential morphisms in $\mathfrak {KNJ}$ are closely related to the essential embeddings in $\mathfrak {KReg}$. We provide a characterization of the $\mathcal {P}$-essential morphisms in $\mathfrak {KNJ}$ and a connection to the essential embeddings in $\mathfrak {KReg}$. Further results about the preservation of joinfitness, the factorization of morphisms, and monomorphisms in $\mathfrak {KNJ}$ are provided. Moreover, in the category of $\mathfrak {KNJ}$ objects and skeletal frame homomorphisms, $\mathfrak {KNJS}$, we construct for $F \in \mathfrak {KNJ}$ and $\phi :\rho F \longrightarrow H$ (an arbitrary $\mathfrak {KReg}$ essential embedding of $\rho F$) the $\mathfrak {KNJS}$ pushout of $\rho _F: \rho F \longrightarrow F$ and $\phi : \rho F \longrightarrow H$. Lastly, we investigate the epimorphisms and epicomplete objects in $\mathfrak {KNJS}$.