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

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-multiorder property, generalizing the order property, and use this to create op-rank, which generalizes 2-rank. From this we build opdimension. We show that op-dimension bounds dp-rank, that opdimension is sub-additive, and op-dimension generalizes o-minimal dimension in o-minimal theories.