Page 68 - Textos de Matemática Vol. 41

54 Chapter 8. Sublocales determined by binary relations
Proposition. Let C be a join-basis of L and let R ⊆ L × L be such that ∀a,b∈L ∀c∈C, aRb ⇒ (a∧c)R(b∧c).
Then s ∈ L is R-saturated iff
aRb ⇒ (a≤s iff b≤s).
If moreover aRb ⇒ a ≤ b this reduces to
aRb ⇒ (a≤s ⇒ b≤s),
or, trivially rewritten, to
aRb & (a≤s) ⇒ b≤s.
Proof. Let c ∈ L and take a join c = i∈J ci with ci ∈ C. Then a ∧ c ≤ s iff
(a∧ci)≤siff∀i,a∧ci ≤siff∀i,b∧ci ≤siff (b∧ci)≤siffb∧c≤s.
i∈J i∈J
8.5. Induced congruences. Let X be a topological space, let A be any of its
subspaces, and let U,V be open sets in X. The relation
UEAV ≡def U∩A=V ∩A (8.5.1)
is the frame congruence (the so-called induced congruence [25]) representing Lc(A) (more precisely, A = jA[Lc(A)], see 5.5) as a sublocale of Lc(X). We will show that an analogous formula, with open sublocales in the role of open subsets, holds for general sublocales of general locales.
8.5.1. Observation. In a distributive lattice let a,b have complements ¬a,¬b. Then for any c,
¬a∧c=¬b∧c iff a∧c=b∧c.
Proof. Indeed let a∧c=b∧c. Then
¬a∧c = (¬a∧b∧c)∨(¬a∧¬b∧c) = (¬a∧a∧c)∨(¬a∧¬b∧c) = ¬a∧¬b∧c,
¬b∧c = (¬b∧a∧c)∨(¬b∧¬a∧c) = (¬b∧b∧c)∨(¬b∧¬a∧c) = ¬a∧¬b∧c. By symmetry we get the other implication.
8.5.2. Proposition. Let E be a frame congruence on a locale L. Let S be the corresponding sublocale. Then
uEv iff o(u)∩S =o(v)∩S. (8.5.2)