Some Contributions on $P_F$-frames

TL;DR

This paper develops the point-free analogue of $P_F$-spaces by introducing and studying $P_F$-frames, showing their place between $P$-frames and $F$-frames and their stability under key frame constructions. It provides several equivalent characterizations via cozero-ideals and multiple ideal-theoretic notions ($z$-, $d$-, $z_r$-ideals) and proves preservation and reflection properties under cozero-onto and nearly open frame morphisms. A central contribution is the ring-theoretic perspective: for normal frames, $L$ is a $P_F$-frame iff $ ext{$\\mathcal{R}$}L$ is $VN$-local, with further equivalences involving $P$-ideals and related ideals in $ ext{$\\mathcal{R}$}L$, thereby linking point-free topology to ideal theory in $LR$. Collectively, these results deepen the understanding of how $P_F$-frames interact with $βL$, cozero elements, and associated ideals, and they extend classical $P_F$-space results to the frame setting.

Abstract

The concept of $P_F$-frames was introduced by Ngoako [24] as a point-free extension of $P_F$-spaces. We observe that the open cozero quotient of a $P_F$-frame is itself a $P_F$-frame. The class of $P_F$-frames contains the class of $P$-frames and is, in turn, contained in the class of $F$-frames. We show that a frame $L$ is a $P_F$-frame if and only if $βL$ is a $P_F$-frame. Moreover, $P_F$-frames are precisely those essential $P$-frames that are also $F$-frames. Lastly, we provide a characterization of $P_F$-frames via $z$-, $d$-, and $z_r$-ideals.

Theorems & Definitions (50)

• Definition 3.1
• Proposition 3.1
• proof
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Corollary 3.1
• Proposition 3.3
• proof