Submission date: 9 May 2022

Abstract:

Given a real closed field R, we identify exactly four proper reducts of R which expand the underlying (unordered) R-vector space structure. Towards this theorem we introduce a new notion, of strongly bounded reducts of linearly ordered structures: A reduct M of a linearly ordered structure < R;<,... > is called strongly bounded if every M-definable subset of R is either bounded or co-bounded in R. We investigate strongly bounded additive reducts of o-minimal structures and as a corollary prove the above theorem on additive reducts of real closed fields.

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Full text arXiv 2205.04199: pdf, ps.