On dp-minimal expansions of the integers II

2024/02/17 by Alouf, Eran
#03C45 #03C65 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2402.11146

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 G⁰⁰≠ G⁰, then for some α∈ℝ\backslashℚ, 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.

Related

• On dp-minimal expansions of the integers
• A family of dp-minimal expansions of (ℤ;+)
• Dp-minimal expansions of discrete ordered abelian groups
• Dp-minimal expansions of (ℤ,+) via dense pairs via Mordell-Lang
• On dp-minimal ordered structures