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58 Chapter 9. Products. Completeness of the category Loc
9.2. Down-sets. Given a poset (X, ≤), the sets A ⊆ (X, ≤) such that ↓A = A are called down-sets or decreasing sets. The set of all down-sets of (X, ≤) ordered by inclusion will be denoted by
D(X, ≤).
Obviously it is a complete lattice with intersections and unions for meets and joins. It satisfies the frame distributive law (in fact it satisfies much more) and hence it is a frame (locale).
We will speak of a U ∈ D(′i∈J Li) as saturated if
∀α∀{xk |k∈K}⊆Lα, {xk∗αu|k∈K}⊆U ⇒ (xk)∗αu∈U.
k
Observation. Consider the relation R on D(′i∈J Li) constituted by all the cou- ples ↓(xk ∗α u),↓(xk)∗α u. (R)
kk
Then R satisfies the two conditions of 8.4 (for the join-basis we can take C = {↓v|v∈ ′i∈JLi}).Sinceforadown-setU,v∈Uiff↓v⊆Uweimmediately conclude that
the above defined saturated down-sets U are precisely the R-saturated elements.
9.3. The product i∈J Li. Define
Li
i∈J
as thesublocale U ⊆ ′i∈J Li | U saturated ofD( ′i∈JLi).
9.3.1. Some particular elements of i∈J Li. The saturation condition did not exclude the void {xk | k ∈ K} (that is, the one with void K). Since ∅ = 0,
every u such that there is an i with ui = 0 is in every saturated set. In other words, set ′
n= u∈ Li|∃i,ui=0. i∈J