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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: pillay@irishmvs.bitnet
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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

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