Borovik-Poizat rank and stability

TL;DR

This paper explores the relationship between Borovik-Poizat rank and stability, demonstrating that ranked structures, despite not being $eth_0$-stable, are in fact superstable, thus connecting rank axioms with stability theory.

Contribution

It proves that structures with Borovik-Poizat rank are superstable, clarifying the connection between rank axioms and stability in model theory.

Findings

Ranked structures are superstable.

Borovik-Poizat rank does not imply $eth_0$-stability.

Connection established between rank axioms and stability theory.

Abstract

There is an axiomatic treatment of Morley rank in groups, due to Borovik and Poizat. These axioms form the basis of the algebraic treatment of groups of finite Morley rank which is common today. There are, however, ranked structures, i.e. structures on which a Borovik-Poizat rank function is defined, which are not $\aleph_0$-stable. Poizat raised the issue of the relationship between this notion of rank and stability theory in the following terms: ``un groupe de Borovik est une structure stable, alors qu'un univers rang\'e n'a aucune raison de l'\^etre ...''. Nonetheless, we show that a ranked structure is superstable.