Page 93 - Textos de Matemática Vol. 41

12.2. Composition of entourages 79
By (2), this join is equal to
{a ⊕ b | x, y, z ̸= 0, x ≤ a, z ≤ b and (x ⊕ y ⊆ E, y ⊕ z ⊆ F ) ⇒ x ⊕ z ⊆ G},
whichispreciselyE◦F →G.
Using Corollary 6.2.3, the collection Ent(L) of all entourages of L may be described by all E ∈ L ⊕ L such that ∆∗(E) = 1, ordered by inclusion. This identification simplifies the writing; for instance, (12.2.1) gives the composite of entourages E and F
E◦F ={a⊕b|(a,c)∈E,(c,b)∈F forsomec̸=0}.
Evidently this definition makes sense for arbitrary E, F ∈ L ⊕ L. In 12.5 below we shall see that ◦ is associative. The operators E ◦ (−) and (−) ◦ E are clearly order-preserving.
12.2.2. The inverse of an entourage E has the natural definition E−1 = r∗(E) where r is the automorphism L ⊕ L → L ⊕ L interchanging the two product projections L ⊕ L → L
L oo p1 L ⊕ L
p2 // L ==
that is,
An entourage E is called symmetric if E−1 = E.
aaC CCCCCC
OO
{{{{{{ {{{
r
p2 CCCC{{{{p1
CCC
C {{
L⊕L
E−1 = {(a,b) | (b,a) ∈ E}.
12.2.3. Proposition. Let a, b ∈ L and E, F ∈ L ⊕ L. Then: (1) (a⊕b)−1 =b⊕a.
(2) (E◦F)−1 =F−1 ◦E−1.
(3) E◦n=n◦E=n.
(4) If E is an entourage then E ⊆ E ◦ E.
i∈J}⊆L⊕L.Ofcourse,E−1 ⊆( Ei)−1 foreveryi∈J.Ontheother i i∈J
Proof. (1) and (3) are obvious.
(2) By (1), it suffices to check that (i∈J Ei)−1 = i∈J(Ei)−1 for every {Ei |