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O-minimal fields with standard part map

100%

Maříková J.

Fundamenta Mathematicae

2010

tom 209

nr 2

115-132

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

Triangulation in o-minimal fields with standard part map

63%

van den Dries L., Maříková J.

Fundamenta Mathematicae

2010

tom 209

nr 2

133-155

In answering questions of J. Maříková [Fund. Math. 209 (2010)] we prove a triangulation result that is of independent interest. In more detail, let R be an o-minimal field with a proper convex subring V, and let st: V → k be the corresponding standard part map. Under a mild assumption on (R,V) we show that a definable set X ⊆ Vⁿ admits a triangulation that induces a triangulation of its standard part st X ⊆ kⁿ.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Maříková J.

1 van den Dries L.

2 2010

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O-minimal fields with standard part map

Triangulation in o-minimal fields with standard part map