Page 92 - Textos de Matemática Vol. 41

78 Chapter 12. Entourages and (quasi-)uniformities
where IE,F is the set
(x, y, z) ∈ L×L×L | x = 0 or y = 0 or z = 0 or (x⊕y ⊆ E, y⊕z ⊆ F, y ̸= 0), together with maps φ1 : M →↓F and φ2 : M →↓E defined by
φ∗1(H)={x⊕y⊕z∈L⊕L⊕L|x⊕y⊆H andy⊕z⊆F}, φ∗2(H)={x⊕y⊕z∈L⊕L⊕L|x⊕y⊆E andy⊕z⊆H}.
(It is very easy to see that φ∗1 and φ∗2 are frame homomorphisms; further
( p ′2 φ 1 ) ∗ ( a ) = φ ∗1 ( ( 1 ⊕ a ) ∩ F ) = { x ⊕ y ⊕ z | x ⊕ y ⊆ E , y ⊕ z ⊆ F , y ≤ a }
and
( p ′1 φ 2 ) ∗ ( a ) = φ ∗2 ( ( a ⊕ 1 ) ∩ E ) = { x ⊕ y ⊕ z | x ⊕ y ⊆ E , y ⊕ z ⊆ F , y ≤ a } .
Thus p′2φ1 = p′1φ2. If ψ1 : N →↓F and ψ2 : N →↓E satisfy p′2ψ1 = p′1ψ2, thereexistsauniqueψ:N→Msuchthatφ1ψ=ψ1 andφ2ψ=ψ2.Indeed, ψ : N → M defined by
ψ∗(G) = {ψ1(x ⊕ y) ∩ ψ2(y ⊕ z) | (x, y, z) ∈ G} is the unique such map, as can be easily checked.)
(2) φ : M → L ⊕ L is defined by
φ∗(G) = {x ⊕ y ⊕ z | x ⊕ y ⊆ E, y ⊕ z ⊆ F, x ⊕ z ⊆ G}
(indeed: φ∗(G) = φ∗((a,b)∈G(a ⊕ b)) = (a,b)∈G(φ∗(a ⊕ 1) ∧ φ∗(1 ⊕ b)) = (a,b)∈G((p′1φ2)∗(a) ∧ (p′2φ1)∗(b)). Then, by (1), we have
φ∗(G) = {x⊕y⊕z|x⊕y⊆E,y⊕z⊆F,x≤a,z≤b,a⊕b⊆G} = {x⊕y⊕z|x⊕y⊆E,y⊕z⊆F,x⊕z⊆G}.)
In particular, φ∗(a⊕b) = {x⊕y⊕z | x⊕y ⊆ E,y⊕z ⊆ F,x ≤ a,z ≤ b}. We are now ready to prove the theorem. For any G ∈ L ⊕ L we have
mm∗(G) = {H∈L⊕L|m∗(H)⊆m∗(G)} = {H ∈L⊕L|φ∗(H)⊆φ∗(G)}
= {a⊕b|φ∗(a⊕b)⊆φ∗(G)}.