Additive reducts of real closed fields and strongly bounded structures

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 the 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 cobounded in R. We investigate strongly bounded additive reducts of o-minimal structures and prove the above theorem on additive reducts of real closed fields.