Trace definability II: model-theoretic linearity

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 $\mathcal{M}$ such that $\mathcal{M}$ does not interpret an infinite group but the Shelah completion of $\mathcal{M}$ 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.

arXiv commentThis is the second in a series of papers consisting of cleaned up and strengthened versions of parts of arXiv:2504.05566v1