John Baldwin
University of Illinois, Chicago

"Expansions of Structures"

ABSTRACT: By an expansion $A^+$ of an $L$-structure $A$, we mean the result of extending the language by adding a new predicate symbol $R$ to $L$, forming $L^+$ and interpreting $R$ on $A$. The study of when expansions preserve such notions as strong minimality, finite rank, o-minimality and stablity has become a common theme in model theory. There are important connections with the abstract theory of data bases (embedded finite model theory), the model theory of fields, and random graph theory. E.g. adding a random relation with edge probability $n^{-\alpha}$ ($\alpha$ irrational) preserves stability. We will survey some of these connections and discuss the following result (joint with Bektur Baizhanov). A subset $A$ of a structure $M$ is {\em benign} if for any $c$, $\tp(c,A)$ implies $\tp_*(c,A)$ (where $*$-types are in the language with a predicate naming $A$). Theorem 1. If $A$ is a uniformly weakly benign subset of a stable structure and the `induced' structure on $A$ is stable then $T^*$ is stable. Further work with Shelah has shown that at least in superstable theories there are many weakly benign subsets.

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