Page 101 - Textos de Matemática Vol. 41

12.8. Localic groups 87 12.7.2. Examples of quasi-uniform locales. For each complemented element a of
a locale L, let
Ea =(a⊕1)∨(1⊕¬a).
Since (a⊕1)∪(1⊕¬a) is already saturated, then Ea = (a⊕1)∪(1⊕¬a). With these entourages one may construct many examples of quasi-uniform locales, with the help of the following result from [29]:
Proposition. Let (L,L1,L2) be a strictly zero-dimensional biframe (i.e. each a ∈ L1 is complemented in L, with complement in L2, and L2 is generated by these complements). The family {Ea | a ∈ L1} forms a subbasis for a quasi- -uniformity on L with induced biframe (L, L1, L2).
For instance, for any locale L, the triple (Sl(L)op, cL, oL)
is a strictly zero-dimensional biframe (recall 6.2.4). Therefore, by the proposi- tion above, {Ec(a) | a ∈ L} is a subbasis for a quasi-uniformity E on Sl(L)op, satisfying L1(E) = cL ≃ L.
This is the point-free counterpart of the well-known fact that every topological space X is quasi-uniformi- zable (in the sense that there exists a quasi-uniformity EX on X admissible with X, that is, which induces as its first topology the given topology Ω(X) of X).
As for topological groups, uniformities are useful for studying localic groups. A localic group is a group object in the category of locales (just as topological groups are the group objects in the category of topo-
logical spaces). Thus a localic group is a locale G equipped with localic maps ε:P→G(theunit), m:G⊕G→G(themultiplication)
and
ı : G → G (the inversion)
subject to the familiar group laws by which the following diagrams commute:
12.8. Localic groups.
G ⊕ G ⊕ G m⊕idG idG⊕m
m
G⊕G G
// G ⊕ G (associativity), m
//