Between groups and rings (Q2277538)

The authors study the system of sets in \({\mathbb{R}}^ n\) definable over the ordered group \({\mathbb{R}}\) equipped with restrictions of linear functions to some bounded interval. In particular, linear functions on all \({\mathbb{R}}\) are not definable over this structure, and the theory of such an expansion of the ordered group \({\mathbb{R}}\) admits elimination of quantifiers. This investigation is inspired by the problem to find a system of sets lying strictly between the system of semilinear sets and the system of semialgebraic sets.