Foundation ranks and supersimplicity

TL;DR

This paper introduces new foundation ranks based on dividing relations, explores their properties, and proposes a concrete definition of supersimple types, showing DU characterizes supersimplicity.

Contribution

It defines the DU rank and a new approach to the D rank, clarifies the concept of supersimple types, and establishes the relationship between these ranks and supersimplicity.

Findings

DU characterizes supersimplicity

D rank does not characterize supersimplicity

New foundation ranks based on dividing relations

Abstract

We introduce a new foundation rank based in the relation of dividing between partial types. We call DU to this rank. We also introduce a new way to define the D rank over formulas as a foundation rank. In this way, SU, DU and D are foundation ranks based in the relation of dividing. We study the properties and the relations between these ranks. Next, we discuss the possible definitions of a supersimple type. This is a concept that it is not clear in the previous literature. In this paper we give solid arguments to set up a concrete definition of this concept and its properties. We also see that DU characterizes supersimplicity, while D not.