Additive reducts of real closed fields and strongly bounded structures

TL;DR

This paper classifies four specific reducts of real closed fields that preserve the underlying vector space structure, introducing strongly bounded reducts to analyze their properties within ordered structures.

Contribution

It introduces the concept of strongly bounded reducts and characterizes the four proper additive reducts of real closed fields.

Findings

Identified exactly four proper reducts of real closed fields.

Introduced the notion of strongly bounded reducts in linearly ordered structures.

Proved the classification of additive reducts of real closed fields.

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 $\langle R;<,\cdots\rangle $ 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.