Page 65 - Textos de Matemática Vol. 41

Chapter 8
Sublocales determined by binary relations
8.1. The sublocale generated by a relation. Although we mostly prefer the covariant description of sublocales as in 5.1, that is, really as sub-locales, subsets the embeddings of which are localic maps, the alternatives as in 5.3.1-5.3.3 are sometimes very useful in constructions.
With a sublocale S ⊆ L we have associated the congruence x ES y iff (∀s ∈ S, x ≤ s iff y ≤ s)
and the nucleus

νS(x) = {s ∈ S | x ≤ s} (8.1.1) (so that xES y iff νS(x) = νS(y)), and the sublocale S can be reconstructed
from the congruence by setting
S={xmax |x∈L}wherexmax =E [x]={y|yE x}.
It often happens that we have given, instead of a congruence, a set R ⊆ L × L of couples to be identified, and we would like to determine the sublocale associated with the congruence generated by R. In fact, it will turn out that we can describe this sublocale more or less directly, without constructing the congruence first.
It should be noted that in frames an explicit extension of a relation to a congruence is not quite such a hard task as in other type of algebras. The fact that each congruence class has a maximal element helps to make it fairly transparent.
ES ES S S
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