Countable models of weakly quasi-o-minimal theories I

Slavko Moconja; Predrag Tanović

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arXiv · 2412.20589

Countable models of weakly quasi-o-minimal theories I

Slavko Moconja·Predrag Tanović

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Abstract

We introduce the notions of triviality and order-triviality for global invariant types in an arbitrary first-order theory and show that they are well behaved in the NIP context. We show that these two notions agree for invariant global extensions of a weakly o-minimal type, in which case we say that the type is trivial. In the o-minimal case, we prove that every definable complete 1-type over a model is trivial. We prove that the triviality has several favorable properties; in particular, it is preserved in nonforking extensions of a weakly o-minimal type and under weak nonorthogonality of weakly o-minimal types. We introduce the notion of a shift in a linearly ordered structure that generalizes the successor function. Then we apply the techniques developed to prove that every weakly quasi-o-minimal theory that admits a definable shift has $2^{\aleph_0}$ countable models.

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BibTeX RIS

Slavko Moconja, Predrag Tanović. 2024-12-29. Countable models of weakly quasi-o-minimal theories I. https://arxiv.org/abs/2412.20589

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