On the class of flat stable theories

TL;DR

This paper introduces flat theories, a subclass of strong stable theories characterized by a new independence relation, showing they have finite weight and group-theoretic properties similar to superstable theories.

Contribution

It defines flat theories via a new independence notion, expanding the class of stable theories with properties akin to superstable theories.

Findings

Flat theories have finite weight for all types.

Type-definable groups in flat theories satisfy chain conditions.

Any type in a flat theory is non-orthogonal to a regular type.

Abstract

A new notion of independence relation is given and associated to it, the class of flat theories, a subclass of strong stable theories including the superstable ones is introduced. More precisely, after introducing this independence relation, flat theories are defined as an appropriate version of superstability. It is shown that in a flat theory every type has finite weight and therefore flat theories are strong. Furthermore, it is shown that under reasonable conditions any type is non-orthogonal to a regular one. Concerning groups in flat theories, it is shown that type-definable groups behave like superstable ones, since they satisfy the same chain condition on definable subgroups and also admit a normal series of definable subgroup with semi-regular quotients.