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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: [email protected]
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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

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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