GALOIS THEORY 3
But yi = (yi)*= Pi- For V^^ = 1 it is enough to show that y. is
infective. For finite I this is clear, and for arbitrary I it is a
consequence of the following:
Proposition 2. Let (M.,a..) be a directed system of sup-lattices.
1
1J
iel
Suppose that for i £ j, a..: M. -* » M. is injective. Then, for every
j e I, the canonical morphism
a.:M. - lim M.
3 J i
iel
is injective.
Proof: By duality it is equivalent to prove that the morphisms
a.
:
lim M. M.
J i J
iel
are surjective. Write a.. = $•-, a. = 3-, and let x e M. . For i _ j ,
put x. = (3-;)*(x). We will be done if we show that if k _ i j , then
6ki(xk) = xi-
xk = (ekj)*(x)
= (Pki)»CBij)*Cx)
= (Bki)»(x.),
and B^-°(3i^-)* = 1 since 3,. is surjective. Thus:
hi^J
= hi°(hO*(xJ
-
xi-
3. Free sup-lattices
As is well-known, the free sup-lattice on a set X in S is the
{ }
power set PX equipped with the singleton map X PX. Every
sup-lattice is a quotient of one of these.
Just as clearly, the free sup-lattice on a partially ordered set
Q is
PQC.PQ,
the set of downward closed subsets of Q, with union as the supremum, and
down-segment +( ):Q - PQ as the universal map.
Since PI is the free sup-lattice on 1, we have
M * Hom(Pl,M).
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