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52 Chapter 8. Sublocales determined by binary relations 8.2. Saturated elements. Let R be a binary relation on a sublocale L. An
element s ∈ S is said to be R-saturated (briefly, saturated) if ∀a,b,c aRb ⇒ (a∧c≤siffb∧c≤s).
The set of all saturated elements will be denoted by
We have
μR =x→{ssaturated |x≤s}:L→L/R.
Define a mapping
L/R.
8.2.1. Proposition. S = L/R is a sublocale of L and μR = jS∗ where jS : S ⊆ L is the embedding map. For the associated nucleus (see (8.1.1)) we have νS (a) = μR(a) and the implication
aRb ⇒ νS(a) = νS(b).
Proof. Obviously L/R is closed under meets. Now let s ∈ L/R. Then for any
x∈L
a∧c≤x→s iff a∧c∧x≤s iff b∧c∧x≤s iff b∧c≤x→s.
Finally, since a ≤ νS(a) ∈ L/R we have b ≤ νS(a) and hence νS(b) ≤ νS(a), and similarly νS(a) ≤ νS(b).
We have μR(x) ≤ s iff x ≤ s = jS(s) for any x ∈ L and s ∈ S and the formula for μR(x) coincides with that in (8.1.1).
8.2.2. Proposition. Let S be a sublocale and let ES be the associated congruence. Then
S = L/ES.
Proof. We have to prove that the ES-saturated elements are precisely the ele-
ments of the form xmax from (8.1). ES
Since ES respects meets (and consequently, of course, the order) we have aE b ⇒ (a∧c)E (b∧c) ⇒ (a∧c≤xmax iffb∧c≤xmax).
S S ES ES
Thus, each xmax is saturated. On the other hand, if s is ES-saturated then we
ES
have in particular, since smax E s and s ≤ s, smax ≤ s. ESS ES