O-minimal fields with standard part map

Jana Maříková¹

¹ Department of Mathematics, WIU 476 Morgan Hall, 1 University Circle Macomb, IL 61455, U.S.A.

Fundamenta Mathematicae, Tome 209 (2010) no. 2, pp. 115-132

Résumé

Let $R$ be an o-minimal field and $V$ a proper convex subring with residue field $\boldsymbol{k}$ and standard part (residue) map $\mathop{\rm st} \colon V\to \boldsymbol{k}$. Let $\boldsymbol{k}_{\rm ind}$ be the expansion of $\boldsymbol{k}$ by the standard parts of the definable relations in $R$. We investigate the definable sets in $\boldsymbol{k}_{\rm ind}$ and conditions on $(R,V)$ which imply o-minimality of $\boldsymbol{k}_{\rm ind}$. We also show that if $R$ is $\omega$-saturated and $V$ is the convex hull of $\mathbb Q$ in $R$, then the sets definable in $\boldsymbol{k}_{\rm ind}$ are exactly the standard parts of the sets definable in $(R,V)$.

Détail

DOI : 10.4064/fm209-2-2

Keywords: o minimal field proper convex subring residue field boldsymbol standard part residue map mathop colon boldsymbol boldsymbol ind expansion boldsymbol standard parts definable relations investigate definable sets boldsymbol ind conditions which imply o minimality boldsymbol ind omega saturated convex hull mathbb sets definable boldsymbol ind exactly standard parts sets definable