Submission date: 4 March 2020

Abstract:

We explore “semibounded” expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We show that if R=< R, <, +, ... > is a semibounded o-minimal structure and P ⊆ R a set satisfying certain tameness conditions, then < R, P > remains semibounded. Examples include the cases when R=ℝ, and P= 2^ℤ, or P=ℤ, or P is an iteration sequence. As an application, we obtain that smooth functions definable in such < R, P > are definable in R.

Mathematics Subject Classification: 03C64, 22B99

Keywords and phrases:

Full text arXiv 2003.02250: pdf, ps.