Complete Congruences on Topologies and Down-set Lattices

Abstract

From the work of Simmons about nuclei in frames it follows that a topological space X is scattered if and only if each congruence on the frame of open sets is induced by a unique subspace A so that = {(U, V) | U ∩ A = V ∩ A}, and that the same holds without the uniqueness requirement iff X is weakly scattered (corrupt). We prove a seemingly similar but substantially different result about quasidiscrete topologies (in which arbitrary intersections of open sets are open): each complete congruence on such a topology is induced by a subspace if and only if the corresponding poset is (order) scattered, i.e. contains no dense chain. More questions concerning relations between frame, complete, spatial, induced and open congruences are discussed as well.

Grothendieck topologies on a poset

“…These results were first presented in [18,Proposition III.4.2]. We also refer to [24, Chapter IX] and [11] for other sources of the material in this section.…”

“…where we used (10) in the first and the last equality, and used (11) in the third and fifth equality. So the diagram indeed commutes.…”

“…Two main sources of this paper are [18] and [11]. In the first, several order-theoretic notions are related to the notion of Grothendieck topologies, whilst in the second, the relation is between these order-theoretic notions and the Artinian property is investigated.…”

Convexities on ordered structures have their Krein--Milman theorem

“…Decomposing elements as joins of coirreducible or coprime elements has been the subject of a great amount of research in order theory (see e.g. Erné [20,21] and references therein, see also Bińczak et al [5,Theorem 5.4] on presentable semilattices), and this theorem invites us to look at these past results from an abstract convexity point of view.…”

An asymmetric characterization of the congruence frame

“…Indeed, Simmons [20] characterizes those spaces for which these are isomorphic as the corrupt or weakly scattered ones. (See also [7].) The congruence lattice of a frame has been studied in various other ways, for instance, using nuclei or sublocales and is also referred to as the assembly or the dissolution locale.…”