Cardinality of the sets of dimension functions in ordered structures

Abstract

We compute the cardinality $\mathfrak n_{\dim}(\mathcal M)$ of the sets of dimension functions on the ordered structures $\mathcal M$. The inequality $\mathfrak n_{\dim}(\mathcal M) \leq 1$ holds if $\mathcal M$ is a d-minimal expansion of an ordered group. If $\mathcal M$ is o-minimal and $\mathfrak n_{\dim}(\mathcal M)<\infty$, there exists a positive integer $m$ such that $\mathfrak n_{\dim}(\mathcal M)=2^m-1$. For every positive integer $m$, there exists a weakly o-minimal expansion $\mathcal M$ of an ordered divisible Abelian group such that $\mathfrak n_{\dim}(\mathcal M)=m$.