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Article · Lobachevskii Journal of Mathematics · December 1, 2023

On o-Stable Expansions of $$\boldsymbol{(\mathbb{Z},<,+)}$$

Viktor Verbovskiy, Aisha Yershigeshova

Viktor Verbovskiy
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Abstract

Roughly speaking, an ordered structure $$(M,<,\dots)$$ is o-stable if any cut in $$M$$ has a few extensions up to complete 1-types over $$M$$ . A theory is o-stable if all its models are. O-stability is a generalization of (weak) o-minimality and quasi-o-minimality by using the notion of a stable theory. In the paper, we prove that no proper (essential) expansion of the ordered group of integers $$(\mathbb{Z},<,+)$$ has an o-stable theory.

Advanced Topology and Set TheoryRings, Modules, and AlgebrasLimits and Structures in Graph Theoryordered groupso-minimalitystable theories

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