On a common generalization of Shelah's 2-rank, dp-rank, and o-minimal dimension

TL;DR

This paper introduces op-dimension, a unified dimension theory that generalizes Shelah's 2-rank, dp-rank, and o-minimal dimension, providing new insights into model-theoretic properties.

Contribution

It defines op-dimension and the n-multi-order property, establishing a common framework that unifies several existing dimension concepts in model theory.

Findings

op-dimension bounds dp-rank

op-dimension is sub-additive

op-dimension generalizes o-minimal dimension

Abstract

In this paper, we build a dimension theory related to Shelah's 2-rank, dp-rank, and o-minimal dimension. We call this dimension op-dimension. We exhibit the notion of the n-multi-order property, generalizing the order property, and use this to create op-rank, which generalizes 2-rank. From this we build op-dimension. We show that op-dimension bounds dp-rank, that op-dimension is sub-additive, and op-dimension generalizes o-minimal dimension in o-minimal theories.