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 6 ki(xk) = x i- x k = (ekj)*(x) = (Pki)»CBij)*Cx) = (Bki)»(x.), and B^-°(3i^-)* = 1 since 3,. is surjective. Thus: hi^J = hi°(hO*(xJ - x i- 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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