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 suplattices.
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 suplattices
As is wellknown, the free suplattice on a set X in S is the
{ }
power set PX equipped with the singleton map X PX. Every
suplattice is a quotient of one of these.
Just as clearly, the free suplattice on a partially ordered set
Q is
PQC.PQ,
the set of downward closed subsets of Q, with union as the supremum, and
downsegment +( ):Q  • PQ as the universal map.
Since PI is the free suplattice on 1, we have
M * Hom(Pl,M).