O-minimal residue fields of o-minimal fields. (English) Zbl 1237.03023
Summary: Let \(R\) be an o-minimal field with a proper convex subring \(V\). We axiomatize the class of all structures \((R,V)\) such that \(\mathbf {k}_{\mathrm{ind}}\), the corresponding residue field with structure induced from \(R\) via the residue map, is o-minimal. More precisely, in [J. Maříková, Fundam. Math. 209, No. 2, 115–132 (2010; Zbl 1221.03029)] it was shown that certain first-order conditions on \((R,V)\) are sufficient for the o-minimality of \(\mathbf {k}_{\mathrm{ind}}\). Here we prove that these conditions are also necessary.
MSC:
| 03C64 | Model theory of ordered structures; o-minimality |
| 03C52 | Properties of classes of models |
| 12L12 | Model theory of fields |
Citations:
Zbl 1221.03029References:
| [1] | Baisalov, Y.; Poizat, B., Paires de structures o-minimales, J. Symbolic Logic, 63, 2, 570-578 (1998) · Zbl 0910.03025 |
| [2] | van den Dries, L., (Tame Topology and o-minimal Structures. Tame Topology and o-minimal Structures, London Mathematical Society Lecture Note Series, vol. 248 (1998), Cambridge University Press) · Zbl 0953.03045 |
| [3] | L. van den Dries, Limit sets in o-minimal structures, in: Proceedings of the RAAG Summer School, O-minimal Structures, Lisbon 2003.; L. van den Dries, Limit sets in o-minimal structures, in: Proceedings of the RAAG Summer School, O-minimal Structures, Lisbon 2003. |
| [4] | van den Dries, L.; Lewenberg, A. H., \(T\)-convexity and tame extensions, J. Symbolic Logic, 60, 74-102 (1995) · Zbl 0856.03028 |
| [5] | van den Dries, L.; Maříková, J., Triangulation in o-minimal fields with standard part map, Fund. Math., 209, 2, 133-155 (2010) · Zbl 1221.03030 |
| [6] | Hrushovski, E.; Peterzil, Y.; Pillay, A., Groups, measures and the NIP, J. Amer. Math. Soc., 21, 563-596 (2008) · Zbl 1134.03024 |
| [7] | Kurdyka, K.; Raby, G., Densité des ensembles sous-analytiques, Ann. Inst. Fourier, 139, 753-771 (1989) · Zbl 0673.32015 |
| [8] | Maříková, J., O-minimal fields with standard part map, Fund. Math., 209, 2, 115-132 (2010) · Zbl 1221.03029 |
| [9] | M. Shiota, PL and differential topology in o-minimal structures, preprint.; M. Shiota, PL and differential topology in o-minimal structures, preprint. |
| [10] | A. Woerheide, O-minimal homology, Ph.D. Thesis, 1996, University of Illinois at Urbana-Champaign.; A. Woerheide, O-minimal homology, Ph.D. Thesis, 1996, University of Illinois at Urbana-Champaign. · Zbl 1153.03010 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
