Strictly zero-dimensional biframes and a characterisation of congruence frames

TL;DR

This paper studies strictly zero-dimensional biframes, showing their categorical completeness and cocompleteness, characterising extremal monomorphisms, and providing an internal characterisation of congruence biframes and frames.

Contribution

It introduces a comprehensive categorical analysis of strictly zero-dimensional biframes and characterises congruence biframes within this framework.

Findings

Category of strictly zero-dimensional biframes is complete and cocomplete.

Characterisation of extremal monomorphisms in this category.

Internal characterisation of congruence biframes and frames.

Abstract

Strictly zero-dimensional biframes were introduced by Banaschewski and Br\"{u}mmer as a class of strongly zero-dimensional biframes including the congruence biframes. We consider the category of strictly zero-dimensional biframes and show it is both complete and cocomplete. We characterise the extremal monomorphisms in this category and explore the special position that congruence biframes hold in it. Finally, we provide an internal characterisation of congruence biframes, and hence, of congruence frames.