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Preprint Number 355

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355. Alf Onshuus, Boris Zilber
The first order theory of universal specializations
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Submission date: 12 September 2011,

Abstract:

This paper concentrates on understanding the first order theory of universal spe- cializations of Zariski structures. Models of the theory are pairs, a Zariski structure and an elementary extension with a map (specialization) from the extension to the structure that preserves positive quantifier free formulas. The reader will find that this context generalizes both the study of algebraically closed valued fields (see [2]) and sheds light on the theory of Zariski structures. It is also a natural setting for studying compact complex manifolds with a standard part map.
We determine the first order theory of universal specializations and prove that it is model complete. Also, we prove that the ground Zariski structure in its core language is stably embedded in models of the theory, which has nice consequences for the theory of Zariski structures.

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