Submission date: 17 February 2024

Abstract:

We first prove that if 𝒵 is a dp-minimal expansion of (ℤ,+,0,1) which is not interdefinable with (ℤ,+,0,1,<), then every infinite subset of ℤ definable in 𝒵 is generic in ℤ. Using this, we prove that if 𝒵 is a dp-minimal expansion of (ℤ,+,0,1) with monster model G such that G^{00} ≠ G^{0}, then for some α ∈ ℝ ∖ ℚ, the cyclic order on ℤ induced by the embedding n ↦ nα+ℤ of ℤ in ℝ/ℤ is definable in 𝒵. The proof employs the Gleason-Yamabe theorem for abelian groups.

Mathematics Subject Classification: 03C45, 03C65

Keywords and phrases:

Full text arXiv 2402.11146: pdf, ps