Page 96 - Textos de Matemática Vol. 41

82 Chapter 12. Entourages and (quasi-)uniformities By Proposition 12.2.3(2), when E is symmetric (that is, E ∈ E implies E−1 ∈ E)
those relations coincide (in that case, we denote them just by E ). 12.3.1 For each entourage E and each a ∈ L let
Ea = {x ∈ L | (x, x) ∈ E, x ∧ a ̸= 0}. Note that E ◦ (a ⊕ a) ⊆ (b ⊕ b) implies Ea ≤ b.
12.3.2. Proposition. Let E be a system of entourages of L. Then:
(1) a≤a′ iE b′ ≤b⇒aiE b.
(2)IfforanyE1,E2 ∈EthereisanF∈EsuchthatF⊆E1∩E2 then a iE b 1 , b 2 ⇒ a iE ( b 1 ∧ b 2 ) a n d a 1 , a 2 iE b ⇒ ( a 1 ∨ a 2 ) iE b .
(3) aiE b⇒a≺b.
Proof. (1) is obvious.
(2)FromE1◦(a⊕a)⊆b1⊕b1 andE2◦(a⊕a)⊆(b2⊕b2)itfollowsthat
(E1 ∩E2)◦(a⊕a) ⊆
⊆ (b1 ⊕b1)∩(b2 ⊕b2)
(E1 ◦(a⊕a))∩(E2 ◦(a⊕a)) ⊆ (b1 ∧b2)⊕(b1 ∧b2).
The other assertion may be proved similarly.
(3) Let E ∈ E such that E ◦ (a ⊕ a) ⊆ (b ⊕ b). Clearly, a ∧ Ea = 0. On the other hand,
Ea∨b ≥ Ea∨{x|(x,x)∈E,x∧a̸=0}
= {x∈L|(x,x)∈E} = ∆∗(E)
= 1.
12.3.3. A system of entourages E of L is said to be admissible if ∀a∈L, a={b∈L|bE a},
where E denotes the filter of (Ent(L), ⊆) generated by E ∪ {E−1 | E ∈ E}. 12.3.4. Proposition. L is regular iff there exists an admissible E on L.