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In response to questions raised by Dolich, Miller, and Steinhorn in their early work on open core, we prove that having an o-minimal open core is not an elementary property. 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In response to questions raised by Dolich, Miller, and Steinhorn in their early work on open core, we prove that having an o-minimal open core is not an elementary property. In particular, we construct an expansion of the structure $(\\mathbb{Q},<)$ that has an o-minimal open core, but some of its elementary superstructures do not."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we prove that having an o-minimal open core is not an elementary property. 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In particular, we construct an expansion of the structure (Q,<) that has an o-minimal open core, but some of its elementary superstructures do not.","one_line_summary":"O-minimality of the open core is not an elementary property, shown via a counterexample expansion of (Q, <) whose elementary superstructures can lack the property.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the specific expansion of (Q,<) can be chosen so its open core is o-minimal while some elementary extension has a non-o-minimal open core; this relies on the construction preserving the necessary first-order properties.","pith_extraction_headline":"Having an o-minimal open core is not an elementary property."},"references":{"count":9,"sample":[{"doi":"","year":2020,"title":"Alexi Block Gorman, Philipp Hieronymi, and Elliot Kaplan, Pairs of Theories Satisfying a Mordell-Lang Condition. Fund. 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