The domination monoid in o-minimal theories

TL;DR

This paper investigates the structure of the domination monoid of invariant types in o-minimal theories, specifically in Real Closed Fields, revealing its generators and computing it in Real Closed Valued Fields.

Contribution

It demonstrates that the domination monoid is generated by classes of 1-types in o-minimal theories and computes it for Real Closed Valued Fields.

Findings

The domination monoid in o-minimal theories is generated by classes of 1-types.

In Real Closed Fields, generators correspond to invariant convex subrings.

The domination monoid in Real Closed Valued Fields is explicitly computed.

Abstract

We study the monoid of global invariant types modulo domination-equivalence in the context of o-minimal theories. We reduce its computation to the problem of proving that it is generated by classes of 1-types. We show this to hold in Real Closed Fields, where generators of this monoid correspond to invariant convex subrings of the monster model. Combined with arxiv:1702.06504, this allows us to compute the domination monoid in the weakly o-minimal theory of Real Closed Valued Fields.