# A special class of congruences on $\kappa$-frames

## TL;DR

This paper explores special congruences on $ppa$-frames, characterizing d-reduced $ppa$-frames via Boolean embeddings and introducing the notion of clear congruences to analyze their structure.

## Contribution

It introduces the concept of clear congruences and characterizes d-reduced $ppa$-frames as quotients by these congruences, extending known Boolean frame results.

## Key findings

- D-reduced $ppa$-frames are quotients by clear congruences.
- Every $ppa$-frame congruence is a meet of clear congruences.
- Characterization of d-reduced $ppa$-frames via Boolean embeddings.

## Abstract

Madden has shown that in contrast to the situation with frames, the smallest dense quotient of a $\kappa$-frame need not be Boolean. We characterise these so-called d-reduced $\kappa$-frames as those which may be embedded as a generating sub-$\kappa$-frame of a Boolean frame. We introduce the notion of the closure of a $\kappa$-frame congruence and call a congruence clear if it is the largest congruence with a given closure. These ideas are used to prove $\kappa$-frame analogues of known results concerning Boolean frame quotients. In particular, we show that d-reduced $\kappa$-frames are precisely the quotients of $\kappa$-frames by clear congruences and that every $\kappa$-frame congruence is the meet of clear congruences.