Supersimplicity and countable reducts of a unidimensional hypersimple theory

TL;DR

This paper proves that a hypersimple unidimensional theory with a large set of countable reducts of finite rank is necessarily supersimple, advancing understanding of the structure of such theories.

Contribution

It establishes a new criterion linking the existence of a club of reducts with finite rank to supersimplicity in hypersimple unidimensional theories.

Findings

Hypersimple unidimensional theories with a club of finite rank reducts are supersimple.

The paper introduces a connection between reducts' properties and the overall theory's simplicity.

It advances the classification of theories based on reducts and rank properties.

Abstract

We show that a hypersimple unidimensional theory that has a club of reducts, in the partial order of all countable reducts, that are coordinatized in finite rank, is supersimple.