Some Contributions on $P_F$-frames

TL;DR

This paper explores properties and characterizations of $P_F$-frames, a class extending $P$-frames, including their behavior under quotients and their relation to $F$-frames, with new characterizations via specific ideals.

Contribution

It provides new characterizations of $P_F$-frames, shows their stability under open cozero quotients, and clarifies their relationship with $P$-frames and $F$-frames.

Findings

Open cozero quotient of a $P_F$-frame is a $P_F$-frame

$L$ is a $P_F$-frame iff $eta L$ is a $P_F$-frame

$P_F$-frames are essential $P$-frames that are also $F$-frames

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 $\beta L$ 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.