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arXiv · 2606.12982

A note on uniform finiteness in weakly o-minimal theories

Yatir Halevi·Assaf Hasson·Ya'acov Peterzil
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Abstract

For an $\aleph_0$-saturated weakly o-minimal expansion of an ordered group $\mathcal{M}$, it is shown that $\mathcal{M}^{\text{eq}}$ has uniform finiteness if and only if the collection of definable convex subgroups of $\mathcal{M}$ has uniform finiteness. If $\mathcal{M}$ expands an ordered field, then considering definable convex valuation subrings is sufficient. The results use a criterion of Johnson for uniform finiteness in $\mathcal{M}^{\text{eq}}$, [8]. In addition, it is shown that uniform finiteness in $\mathcal{M}^{\text{eq}}$ may fail for weakly o-minimal expansions of fields.

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Yatir Halevi, Assaf Hasson, Ya'acov Peterzil. 2026-06-11. A note on uniform finiteness in weakly o-minimal theories. https://arxiv.org/abs/2606.12982

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