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 the new notion of strongly bounded reducts of linearly ordered structures: a reduct $\mathcal{ℳ}$ of a linearly ordered structure $⟨R;<,\dots <!--FUNCTION\; APPLICATION-->⟩$ is called strongly bounded if every $\mathcal{ℳ}$-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.

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