Download Existentially Closed Groups by Graham Higman, Elizabeth Scott PDF

By Graham Higman, Elizabeth Scott

During this quantity, the authors introduce the speculation of existentially closed teams, bringing jointly either well-established and extra modern rules, interpretations, and proofs. They undertake a group-theoretical instead of a model-theoretical perspective as they outline existentially closed teams and summarizes the various suggestions which are easy to countless workforce idea, resembling the formation of loose items with amalgamation and HNN-extensions. From this foundation the idea is built and plenty of of the extra lately found effects are proved and mentioned.

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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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