Dense-codense expansions of quasiminimal pregeometry structures

Alexander Berenstein; Evgueni Vassiliev

Dense-codense expansions of quasiminimal pregeometry structures

Logic 2026-07-13 v1

Authors: Alexander Berenstein , Evgueni Vassiliev

Abstract

We study expansions of quasiminimal pregeometry structures with a dense codense unary predicate and their relation with the complexity properties of the pregeometry of the underlying structure. We consider beautiful pairs as well as $H$ -structures. We show each of these expansions can be axiomatized with a single $L_{\omega_1 \omega}(Q)$ -sentence and that both expansions are $\omega$ -stable. For $H$ -structures we provide a natural notion of independence in the expansion and when the underlying structure is modular, we also provide a natural notion of independence for beautiful pairs. Then we relate the complexity of the pregeometry to properties of the expansions.

Cite

@article{arxiv.2607.11619,
  title  = {Dense-codense expansions of quasiminimal pregeometry structures},
  author = {Alexander Berenstein and Evgueni Vassiliev},
  journal= {arXiv preprint arXiv:2607.11619},
  year   = {2026}
}