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By Julia Driver

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Ws) ( E W(x) ) . • . • • • • • • • . • • • . for the result of replacing each occurrence of y; or y i 1 in z by w; or w7, respectively. Define f : W(y) � W(x) by the rule : f (uj ) = uj (W1 , . . Ws)vjvj for j"� 1 (we can, and we will, assume that U o and v0 are the empty words of W(y) and W(x), respectively, and that f (uo) = v o) · Then f is recursive and, since ulb 1 , . . , bs) = ui (w1 (a), . . , ws (a)) = ui (w1 (a), . . , ws (a))vj(a)vla), (where w;(a) = w;(a 1 , . . , a,)), it is clear that ui E Rel (b 1 , , bs) if and only if f (ui) E Rel (a 1 , .

Then X is countable, Sk X = gr, and if G � X is a finitely generated group, if k E X, and if () : G ----;. X is any monomorphism, then there exists i such that G, h, and G () all lie in X;. By construction, there exists a monomorphism {) : ( G, h ) ----;. X; + r . which extends () : G ----;. X;. Thus, X is w-homogeneous D and we have completed stage III . 9 If gr is a non-trivial non-empty class of finitely generated groups that consists of at most a countable set of distinct isomorphism types, then gr is the skeleton of a countable existentially closed group if and only if it satisfies SC, JEP, and AC.

D Notice that it cannot be true that, for every X £ N, there exists a finitely generated group G such that X = 1 Rel G, since Rel G cannot be finite. 13 An ideal l is a non-empty subset of an upper semi­ lattice L such that: ( i ) If A E l and B E L with B � A, then B E l. ( ii ) If A , B E l then the least upper bound A v B of A and B lies in l. P(N) under enumeration reducibility, is an upper semilattice . P(N)/=e generated by X. P(N)/=e which are not of the form J(X), for any X £ N. P(N) is uncountable ) some Z £ N such that Z fe Y.

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