Semilattice base hierarchy for frames and its topological ramifications
Abstract
We develop a hierarchy of semilattice bases (S-bases) for frames. For a given (unbounded) meet-semilattice , we analyze the interval in the coframe of sublocales of the frame of downsets of formed by all frames with the S-base . We give an explicit description of the nuclei associated with these sublocales. We study various degrees of completeness of , 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.
Keywords
Cite
@article{arxiv.2308.01627,
title = {Semilattice base hierarchy for frames and its topological ramifications},
author = {G. Bezhanishvili and F. Dashiell and A. Razafindrakoto and J. Walters-Wayland},
journal= {arXiv preprint arXiv:2308.01627},
year = {2024}
}