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Complete congruences on topologies and down-set lattices. (English) Zbl 1122.06015

Summary: From the work of Simmons about nuclei in frames it follows that a topological space \(X\) is scattered if and only if each congruence \(\Theta\) on the frame of open sets is induced by a unique subspace \(A\) so that \(\Theta = \{(U,V)\mid U\cap A = V \cap A\}\), 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.

MSC:

06F30 Ordered topological structures
06D22 Frames, locales
54H10 Topological representations of algebraic systems
06B10 Lattice ideals, congruence relations
54G12 Scattered spaces
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