Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-26T17:34:50.163Z Has data issue: false hasContentIssue false
  • English
  • Français

Additive reducts of real closed fields

Published online by Cambridge University Press:  12 March 2014

David Marker ,
Ya'acov Peterzil  and
Anand Pillay
David Marker
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60680
Ya'acov Peterzil
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720
Anand Pillay
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556, E-mail: pillay@irishmvs.bitnet
Get access Rights & Permissions [Opens in a new window]

Extract

In [MP] Marker and Pillay showed that if X ⊂ Cn is constructible but (C, +, X) is not locally modular, then multiplication is definable in the structure (C, +,X). That result extended earlier results of Martin [M] and Rabinovich and Zil'ber [RZ]. Here we will examine additive reducts of R and Qp.

Definition. A subset X of Rn is called semialgebraic if it is definable in the structure (R, +,·). A subset X of Rn is called semilinear if it is definable in the structure (R, +, <,λr)r∈b, where λr is the function x ↦ rx [scalar multiplication by r].

Every semilinear set is a Boolean combination of sets of the form {: p () = 0} and {: q() > 0}, where p() and q() are linear polynomials.

Van den Dries asked the following question: if X is semialgebraic but not semilinear, can we define multiplication in (R, +, <,X)? This was answered negatively by Pillay, Scowcroft and Steinhorn.

Theorem 1.1 [PSS]. Suppose X ⊂ Rnis semialgebraic and X ⊂ Infor some bounded interval I. Then multiplication is not definable in (R, +, <,X,λr)r∈R.

In particular if X = · ∣ [0, l ]2, the graph of multiplication restricted to the unit interval, then X is not semilinear so we have a negative answer to van den Dries' question. Peterzil showed that this is the only restriction.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 57 , Issue 1 , March 1992 , pp. 109 - 117
Copyright
Copyright © Association for Symbolic Logic 1992

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.)

References

REFERENCES

[BCR]Bochnak, J., Coste, M., and Roy, M.-F., Géométrie algébrique réelle, Springer-Verlag, Berlin, 1987.Google Scholar
[KPS]Knight, J., Pillay, A., and Steinhorn, C., Definable sets in ordered structures. II, Transactions of the American Mathematical Society, vol. 295 (1986), pp. 593–605.CrossRefGoogle Scholar
[Mac]Macintyre, A., On definable subsets of p-adic fields, this Journal, vol. 41 (1976), pp. 605–610.Google Scholar
[MP]Marker, D. and Pillay, A., Reducts of (C, +, ·) that contain +, this Journal, vol. 55 (1990), pp. 1243–1251.Google Scholar
[M]Martin, G., Definability in reducts of algebraically closed fields, this Journal, vol. 53 (1988), pp. 189–199.Google Scholar
[NP]Nesin, A. and Pillay, A., Some model theory for compact Lie groups, Transactions of the American Mathematical Society, vol. 326 (1991), pp. 453–463.CrossRefGoogle Scholar
[Pe]Peterzil, K., Ph.D. thesis, University of California, Berkeley, California, 1991.Google Scholar
[Pi]Pillay, A., On fields definable in Qp, Archive for Mathematical Logic, vol. 29 (1989), pp. 1–7.CrossRefGoogle Scholar
[PSS]Pillay, A., Scowcroft, P., and Steinhorn, C., Between groups and rings, Rocky Mountain Journal of Mathematics, vol. 19 (1989), pp. 871–885.CrossRefGoogle Scholar
[PS]Pillay, A. and Steinhorn, C., Definable sets in ordered structures. I, Transactions of the American Mathematical Society, vol. 295 (1986), pp. 565–592.CrossRefGoogle Scholar
[RZ]Rabinovich, E. and Zil'ber, B., Additive reducts of algebraically closed fields, preprint. (Russian)Google Scholar
[SvdD]Scowcroft, P. and van den Dries, L., On the structure of semialgebraic sets over p-adic fields, this Journal, vol. 53 (1988), pp. 1138–1164.Google Scholar
[ZS]Zariski, O. and Samuel, P., Commutative algebra, Vol. II, Springer-Verlag, Berlin, 1975.Google Scholar