Submission date: 15 April 2020

Abstract:

This is a contribution to the classification problem for dp-minimal expansions of (ℤ,+). Let S be a dense cyclic group order on (ℤ,+). We use results on “dense pairs” to construct uncountably many dp-minimal expansions of (ℤ,+,S). These constructions are applications of the Mordell-Lang conjecture and are the first examples of “non-modular” dp-minimal expansions of (ℤ,+). We canonically associate an o-minimal expansion R of (ℝ,+,×), an R-definable circle group ℍ, and a character ℤ → ℍ to a “non-modular” dp-minimal expansion of (ℤ,+,S). We also construct a “non-modular” dp-minimal expansion of (ℤ,+,Val_p) from the character ℤ → ℤ_p^×, k ↦ exp(pk).

Mathematics Subject Classification:

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Full text arXiv 2004.06847: pdf, ps.