Dense-codense expansions of quasiminimal pregeometry structures
arXiv:2607.11619
The paper investigates model‑theoretic expansions of quasiminimal pregeometry structures by adding a dense codense unary predicate, focusing on beautiful pairs and H‑structures, and shows these expansions are axiomatizable by a single L_{ω1ω}(Q) sentence and are ω‑stable, providing natural independence notions and linking pregeometry complexity to expansion properties.
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 -structures. We show each of these expansions can be axiomatized with a single -sentence and that both expansions are -stable. For -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.