Spectral spaces versus distributive lattices: a dictionary
Abstract
The category of distributive lattices is, in classical mathematics, antiequivalent to the category of spectral spaces. We give here some examples and a short dictionary for this antiequivalence. We propose a translation of several abstract theorems (in classical mathematics) into constructive ones, even in the case where points of a spectral space have no clear constructive content. La cat\'egorie des treillis distributifs et celle des espaces spectraux sont anti\'equivalentes (en math\'ematiques classiques). Nous proposons ici un petit dictionnaire pour cette anti\'equivalence. Nous indiquons comment un certain nombre de th\'eor\`emes \'etranges des math\'ematiques classiques obtiennent un contenu constructif gr\^ace \`a cette anti\'equivalence, m\^eme dans le cas, fr\'equent, o\`u les points des espaces spectraux consid\'er\'es n'ont pas de contenu constructif clair.
Keywords
Cite
@article{arxiv.1812.06277,
title = {Spectral spaces versus distributive lattices: a dictionary},
author = {Henri Lombardi},
journal= {arXiv preprint arXiv:1812.06277},
year = {2025}
}
Comments
small typos have been fixed, references are improved, a french translation has been added. English version appeared p. 223--245 in: Advances in rings, modules and factorizations. Selected papers based on the presentations at the international conference on rings and factorizations, Graz, Austria, February 19--23, 2018. Cham: Springer (2020). ISBN 978-3-030-43415-1