John Goodrick : Bi-embeddability and isomorphism: the weakly minimal case.

Posted by , part of the Logic Seminar.

At

Nov. 3, 2009, 4 p.m.

In

SEO 612

Abstract

Recently (in joint work with Chris Laskowski) we characterized countable, weakly minimal theories $T$ such that any two (elementarily) bi-embeddable models of $T$ are isomorphic. We prove that if $T$ is countable and weakly minimal, the following are equivalent: 1. $T$ has two bi-embeddable but nonisomorphic models; 2. There is an automorphism $f$ of the monster model of $T$ and a strong type $p$ over the empty set which is almost-orthogonal to $f(p) \otimes \ldots \otimes f^n(p) $ for any n. 3. $T$ has an infinite collection of models that are pairwise bi-embeddable but pairwise nonisomorphic. The proof involves some geometric stability theory plus a Dushnik-Miller type argument to build nonisomorphic models by "killing'' every potential isomorphism at each stage of the construction.

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