Functorial Polar Functions in Compact Normal Joinfit Frames

期刊论文
发表日期：

2024年8月23日

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Ricardo E. Carrera

摘要

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