Additive reducts of real closed fields and strongly bounded structures

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 $\mathcal{M}$ of a linearly ordered structure $⟨R;<,\cdots ⟩$ is called \emph{strongly bounded} if every $\mathcal{M}$-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.

Cited by in corpus (1)

• Very ampleness in strongly minimal sets