Submission date: 24 October 2021

Abstract:

We explore semibounded expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We introduce the notion of a semibounded expansion of an arbitrary ordered group, extending the usual notion from the o-minimal setting. For R=( ℝ, <, +, ...), a semibounded o-minimal structure and P ⊆ ℝ a set satisfying certain tameness conditions, we discuss under which conditions (R,P) defines total linear functions that are not definable in R. Examples of such structures that does define new total linear functions include the cases when R is a reduct of (ℝ,<,+,⋅_{↾ (0,1)^2},(x ↦ λ x)_{λ ∈ I ⊆ ℝ), and P= 2^ℤ, or P is an iteration sequence (for any I) or P=ℤ, for I=ℚ.

Mathematics Subject Classification:

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Full text arXiv 2110.12407: pdf, ps.