Semilattice base hierarchy for frames and its topological ramifications

TL;DR

This paper introduces a hierarchy of semilattice bases for frames, analyzing their properties and implications for various classes of topological frames, providing new insights into their structure and relationships.

Contribution

It develops a hierarchy of semilattice bases for frames, introduces D- and L-bases, and offers a detailed analysis of their topological and algebraic properties.

Findings

Explicit description of nuclei associated with sublocales

Generalization of extremally and basically disconnected frames

Application to zero-dimensional, regular, and coherent frames

Abstract

We develop a hierarchy of semilattice bases (S-bases) for frames. For a given (unbounded) meet-semilattice $A$, we analyze the interval in the coframe of sublocales of the frame of downsets of $A$ formed by all frames with the S-base $A$. We give an explicit description of the nuclei associated with these sublocales. We study various degrees of completeness of $A$, which generalize the concepts of extremally disconnected and basically disconnected frames. We also introduce the concepts of D-bases and L-bases, as well as their bounded counterparts, and show how our results specialize and sharpen in these cases. Classic examples that are covered by our approach include zero-dimensional, completely regular, and coherent frames, allowing us to provide a new perspective on these well-studied classes of frames, as well as their spatial counterparts.