Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

Streszczenie

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 $(R,V)$.