UNIFORMLY EXTERNALLY DEFINABLE EXPANSION

In this article, we study the expansion of a structure by adding a new predicate that is not definable by any formula in the original language. To consider an externally definable expansion, we define the extension of a model in both essential and non-essential case. Such expansions can lead to significant changes in the properties of the resulting structure. We focus on the case of externally definable expansions, where the new relation is given by the intersection of a formula defined in an elementary extension with the original structure. The concept of a uniformly externally definable expansion was first introduced by Macpherson, Marker, and Steinhorn in the context of expansions by cuts in submodels of o-minimal structures over the real numbers. Subsequently, Baizhanov demonstrated that expanding a model of a weakly o-minimal theory by a family of convex sets preserves both weak o-minimality and uniform external definability.  We establish conditions for external expansions under which the key properties of the original structure are preserved.

1. Shelah S. Classification theory and the number of non-isomorphic models, Studies in Logic and the Foundations of Mathematics, 1978, 2nd edition 1990, Elsevier, ISBN 978-0-444-70260-9.

2. Macpherson, D., Marker, D., Steinhorn, Ch. Weakly o-minimal structures and real closed fields, Translations of the American Mathematical Society, 352, 5435–5483 (2000).

3. Baizhanov, B. Expansion of a model of a weakly o-minimal theory by a family of unary predicates, The Journal of Symbolic Logic, 66, 1382–1414 (2001).

4. Shelah, S. Dependent first order theories. Continued, Israel Journal of Mathematics 173, 1 (2009). https://doi.org/10.1007/s11856-009-0082-1.

5. Baizhanov, B.S.,Verbovskii, V.V. O-Stable Theories. Algebra and Logic, 50, 211–225 (2011).