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1
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Dense pairs of o-minimal structures

100%
van den Dries L.
Fundamenta Mathematicae
|
1998
|
tom 157
|
nr 1
61-78
EN
The structure of definable sets and maps in dense elementary pairs of o-minimal expansions of ordered abelian groups is described. It turns out that a certain notion of "small definable set" plays a special role in this description.
2
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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
EN
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.
3
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Erratum to "Fields of surreal numbers and exponentiation" (Fund. Math. 167 (2001), 173-188)

63%
van den Dries L., Ehrlich P.
Fundamenta Mathematicae
|
2001
|
tom 168
|
nr 3
295-297
4
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Fields of surreal numbers and exponentiation

63%
van den Dries L., Ehrlich P.
Fundamenta Mathematicae
|
2001
|
tom 167
|
nr 2
173-188
EN
We show that Conway's field of surreal numbers with its natural exponential function has the same elementary properties as the exponential field of real numbers. We obtain ordinal bounds on the length of products, reciprocals, exponentials and logarithms of surreal numbers in terms of the lengths of their inputs. It follows that the set of surreal numbers of length less than a given ordinal is a subfield of the field of all surreal numbers if and only if this ordinal is an ε-number. In that case, this field is even closed under surreal exponentiation, and is an elementary extension of the real exponential field.
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