Borovik-Poizat rank and stability

Extract

Borovik proposed an axiomatic treatment of Morley rank in groups, later modified by Poizat, who showed that in the context of groups the resulting notion of rank provides a characterization of groups of finite Morley rank [2]. (This result makes use of ideas of Lascar, which it encapsulates in a neat way.) These axioms form the basis of the algebraic treatment of groups of finite Morley rank undertaken in [1].

There are, however, ranked structures, i.e., structures on which a Borovik-Poizat rank function is defined, which are not ℵ₀-stable [1, p. 376]. In [2, p. 9] 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 rangé n'a aucune raison de l'être …” (emphasis added). Nonetheless, we will prove the following:

Theorem 1.1. A ranked structure is superstable.

An example of a non-ℵ₀-stable structure with Borovik-Poizat rank 2 is given in [1, p. 376]. Furthermore, it appears that this example can be modified in a straightforward way to give ℵ₀-stable structures of Borovik-Poizat rank 2 in which the Morley rank is any countable ordinal (which would refute a claim of [1, p. 373, proof of C.4]). We have not checked the details. This does not leave much room for strenghthenings of our theorem. On the other hand, the proof of Theorem 1.1 does give a finite bound for the heights of certain trees of definable sets related to unsuperstability, as we will see in Section 5.