Cardinality of the sets of dimension functions in ordered structures

TL;DR

This paper investigates the number of possible dimension functions in various ordered structures, revealing bounds and exact counts depending on the structure's properties, such as o-minimality and weak o-minimality.

Contribution

It provides a detailed computation of the cardinality of dimension functions in ordered structures, establishing bounds and existence results for different classes.

Findings

If $ ext{d-minimal}$, then $ ext{cardinality} o 1$

For o-minimal structures with finite cardinality, it equals $2^m - 1$ for some $m$

For every positive integer $m$, there exists a weakly o-minimal structure with cardinality $m$

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 $\mathfrak n_{\dim}(\mathcal M)=m$.