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80 Chapter 12. Entourages and (quasi-)uniformities hand,ifE−1 ⊆F foreveryi∈J thenEi ⊆F−1 andthus Ei ⊆F−1,that
i i∈J is,( i∈JEi)−1⊆F.
(4)Let(a,b)∈E.Thecaseb=0isobvious.Ifb̸=0,thenb= {b∧c|(c,c)∈ E, b ∧ c ̸= 0} and it suffices to check that (a, b ∧ c) ∈ E ◦ E for each such (c, c). This is obvious, since (a, b ∧ c) ∈ E, (b ∧ c, b ∧ c) ∈ E and b ∧ c ̸= 0.
12.2.4. More about the composition. For any A, B ∈ D(L × L) define the down- -set A · B as
A · B = {(a, b) | (a, c) ∈ A, (c, b) ∈ B for some c ̸= 0}.
This defines an associative operator on D(L × L), which endows D(L×L) with a structure of a quantale (that is, a complete lattice with an associative binary multiplication which distributes over arbitrary sup- rema). For information on quantales consult the monograph [59].
Themapj0 :D(L×L)→D(L×L)givenby
j0(A) = x, S | {x} × S ⊆ A ∪ S, y | S × {y} ⊆ A
is a prenucleus, that is, for every A, B ∈ D(L × L), (1) A ⊆ j0(A),
(2) j0(A)∩B⊆j0(A∩B),and
(3) j0(A) ⊆ j0(B) whenever A ⊆ B.
Consequently,
Fix(j0) := {A ∈ D(L × L) | j0(A) = A} = L ⊕ L
is a closure system, and the associated closure operator is given by
j(A) = {B ∈ L ⊕ L | A ⊆ B}, which is the saturated element of L ⊕ L generated by A.
12.2.5. Lemma. For any A, B ∈ D(L × L), j(A) ◦ j(B) = j(A · B).
Proof. The inclusion j(A · B) ⊆ j(A) ◦ j(B) is obvious. It remains to show that
j(A) · j(B) ⊆ j(A · B). For this, consider the non-empty set
A = {E ∈ D(L × L) | A ⊆ E ⊆ j(A), E · B ⊆ j(A · B)}