Supersimple theories

Abstract

We prove elimination of hyperimaginaries in supersimple theories. This means that if an equivalence relation on the set of realisations of a complete type (in a supersimple theory) is defined by a possibly infinite conjunction of first order formulas, then it is the intersection of definable equivalence relations.

Modular types in some supersimple theories

“…So Tr A (ip{x, b)) c acl(q). By Lemma 3.2 (2), q e acl(r), so we are done. H Forking in simple theories means decreasing some local rank D (-,A,k).…”

“…Then the types stp(b'/Aa) are all distinct extensions of tp{b/Aa), hence Tr Aa (b/Aa) is infinite, a contradiction. (2) follows from Remark 2.1 (2). H In our definition of a modular regular type in a simple theory we follow [6].…”

“…Since we work in a supersimple theory, due to the elimination of hyperimaginaries in this case [2] we can work with imaginaries and strong types instead of hyperimaginaries and Lascar strong types. §2.…”

Reducts of stable, CM-trivial theories

“…n ACL(a, y) (= ACL(Z)) a n d l = I n C e t ' -. If we assume for a contradiction that a J^ y then (1) aJJ^y by Theorem 2.2 since A' is algebraically closed in 7 1 by Theorem 2.2.…”

“…-\ For the rest of this section, let T be a supersimple theory of finite SU-rank. Then T has elimination of hyperimaginaries [1]. We will show now that we can then assume that there are rank 1 elements in a 2-ample (1-ample respectively) tuple.…”