### Model Companions of $T_{\rm Aut}$ for stable $T$

by Baldwin and Shelah. [BlSh:759]
Notre Dame J Formal Logic, 2001
Let T be a complete first order theory in a countable relational language L . We assume relation symbols have been added to make each formula equivalent to a predicate. Adjoin a new unary function symbol sigma to obtain the language L_sigma ; T_sigma is obtained by adding axioms asserting that sigma is an L-automorphism. We provide necessary and sufficient conditions for T_Aut to have a model companion when T is stable. Namely, we introduce a new condition: T admits obstructions, and show that T_Aut has a model companion iff and only if T does not admit obstructions. This condition is weakening of the finite cover property: if a stable theory T has the finite cover property then T admits obstructions.

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