Hostname: page-component-cb9f654ff-pvkqz Total loading time: 0 Render date: 2025-08-11T05:05:31.677Z Has data issue: false hasContentIssue false
  • English
  • Français

CONVEXLY ORDERABLE GROUPS AND VALUED FIELDS

Published online by Cambridge University Press:  17 April 2014

JOSEPH FLENNER  and
VINCENT GUINGONA
JOSEPH FLENNER
Affiliation:
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF SAINT FRANCIS, 2701 SPRING STREET, FORT WAYNE, IN 46808, USA E-mail: jflenner@sf.edu
VINCENT GUINGONA
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NOTRE DAME 255 HURLEY HALL NOTRE DAME IN 46556 USA E-mail: guingona.1@nd.edu, URL: http://www.nd.edu/∼vguingon/
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the model theoretic notion of convex orderability, which fitsstrictly between the notions of VC-minimality and dp-minimality. In some classesof algebraic theories, however, we show that convex orderability andVC-minimality are equivalent, and use this to give a complete classification ofVC-minimal theories of ordered groups and abelian groups. Consequences forfields are also considered, including a necessary condition for a theory ofvalued fields to be quasi-VC-minimal. For example, the p-adicsare not quasi-VC-minimal.

Keywords

Convexly orderableVC-minimalityordered groupsvalued fieldsabelian groups

Information

Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 1 , March 2014 , pp. 154 - 170
Copyright
Copyright © Association for Symbolic Logic 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

Andrews, Uri, Cotter, Sarah, Freitag, James, and Medvedev, Alice, VC-minimality: Examples and observations, in preparation.Google Scholar
Adler, Hans, Theories controlled by formulas of Vapnik-Chervonenkis codimension 1, preprint (2008).Google Scholar
Aschenbrenner, M., Dolich, A., Haskell, D., MacPherson, D., and Starchenko, S., Vapnik- Chervonenkis density in some theories without the independence property, II. Notre Dame Journal of Formal Logic, vol. 54 (2013), no. 3-4, pp. 311–363.Google Scholar
Cotter, Sarah and Starchenko, Sergei, Forking in VC-minimal theories. this Journal, vol. 75 (2012), no. 4, pp. 1257–1271.Google Scholar
Dickmann, M. A., Elimination of quantifiers for ordered valuation rings. this Journal, vol. 52 (1987), no. 1, pp. 116–128.Google Scholar
Dolich, Alfred, Goodrick, John, and Lippel, David, Dp-minimality: Basic facts and examples. Notre Dame Journal of Formal Logic, vol. 52 (2011), no. 3, pp. 267–288.Google Scholar
Flenner, Joseph and Guingona, Vincent, Canonical forests in directed families. Proceedings of the American Mathematical Society, (to appear).Google Scholar
Guingona, Vincent and Laskowski, M. C., On VC-minimal theories and variants. Archive for Mathematical Logic, vol. 52 (2013), no. 7, 743–758.Google Scholar
Holly, Jan E., Canonical forms for definable subsets of algebraically closed and real closed valued fields. this Journal, vol. 60 (1995), no. 3, pp. 843–860.Google Scholar
MacPherson, Dugald, Marker, David, and Steinhorn, Charles, Weakly o-minimal structures and real closed fields. Transactions of the American Mathematical Society, vol. 352 (2000), no. 12, pp. 5435–5483.Google Scholar
Pillay, Anand and Steinhorn, Charles, Definable sets in ordered structures I. Transactions of the American Mathematical Society, vol. 295 (1986), no. 2, pp. 565–592.Google Scholar
Prest, Mike, Model theory and modules, London Mathematical Society Lecture Note Series, vol. 130, Cambridge University Press, Cambridge, 1988.Google Scholar
Simon, Pierre, On dp-minimal ordered structures. this Journal, vol. 76 (2011), pp. 448–460.Google Scholar
Szmielew, Wanda, Elementary properties of Abelian groups. Fundamenta Mathematicae, vol. 41 (1955), pp. 203–271.Google Scholar