Trace definability II: model-theoretic linearity

TL;DR

This paper explores how algebraic structures can emerge in model-theoretic expansions, introducing new notions of interpretability and revealing a dichotomy between linearity and field structures in certain ordered groups.

Contribution

It introduces the concepts of local trace definability and local trace equivalence, and demonstrates a dichotomy between linear and field structures in dp-minimal expansions.

Findings

Constructed a weakly o-minimal structure whose Shelah completion interprets an infinite field.

Established a dichotomy between linearity and field structure in dp-minimal expansions of archimedean ordered abelian groups.

Proved several results on trace definability and local trace definability across various classes of structures.

Abstract

We give examples of $\mathrm{NIP}$ structures in which new algebraic structure appears in the Shelah completion. In particular we construct a weakly o-minimal structure $\mathscr{M}$ such that $\mathscr{M}$ does not interpret an infinite group but the Shelah completion of $\mathscr{M}$ interprets an infinite field. We introduce a weak notion of interpretability called local trace definability between first order structures and an associated weak notion of equivalence. We give a dichotomy between ``linearity" and ``field structure" for dp-minimal expansions of archimedean ordered abelian groups. We also prove several other results about trace definability and local trace definability between various classes of structures.