Page 105 - Textos de Matemática Vol. 41

12.10. Uniformities on localic groups 91 Applying idG ⊕ ∆∗(idG ⊕ ı∗) ⊕ idG to both sides we get
a⊕(b∧ı∗(c))⊕d⊆(idG ⊕σ∗ ⊕idG)(m∗(x))= (a⊕idG ⊕b). (a,b)∈m∗ (x)
Since b ∧ ı∗(c) ̸= 0, then (a, d) ∈ m∗(x) and (ı∗(a), d) ∈ E(x). Hence E(y) ◦ E(y) ⊆ E(x).
Finally, let us check the admissibility condition. From the identity (idG ⊕ ε∗)m∗ = idG
it follows that, for every x ∈ G,
x = {a⊕ε∗(b)|a⊕b⊆m∗(x)}
= {a∈G|∃b∈N :a⊕b⊆m∗(x)},
whence it remains to show that a E x whenever there is some b ∈ N satisfying a ⊕ b ⊆ m∗(x). By Lemma 12.2.2(3), just take a symmetric F ∈ E such that F2 ⊆E(b).
12.10.4. E is called the left uniformity of G. Similarly, the sets F(x)=(ı⊕idG)∗m∗(x), x∈N,
form a basis for a uniformity F, called the right uniformity of G. The two-sided uniformity is the infimum E ∧ F .
In general, the three uniformities are different; however (ı ⊕ ı)∗[E(x)] = F (ı∗(x)) = F (x)−1
and hence ı is a uniform isomorphism from G with its left uniformity to G with its right uniformity.
In the commutative case, where mr = m, the three uniformities coincide because a ⊕ b ≤ m∗(x) iff b ⊕ a ≤ m∗(x). Indeed,
a ⊕ b ≤ m∗(x) ⇒ r∗(a ⊕ b) ≤ r∗m∗(x) = m∗(x), that is b ⊕ a ≤ m∗(x); therefore
E(x)−1 = {(a ⊕ ı∗(b))−1 | a ⊕ b ≤ m∗(x)} = {ı∗(b)⊕a|b⊕a≤m∗(x)}
= F(x).
12.10.5. Any group homomorphism f : G → H is uniform. Indeed, (f ⊕ f)∗(idH ⊕ ıH)∗m∗H(x) = (idG ⊕ ıG)∗m∗G(f∗(x))
and ε∗Gf∗(x) = ε∗H(x) = 1.