A note on uniform finiteness in weakly o-minimal theories

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.