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Dedekind Completeness and the Algebraic Complexity of o-Minimal Structures

Published online by Cambridge University Press:  20 November 2018

Alan Mekler ,
Matatyahu Rubin  and
Charles Steinhorn
Alan Mekler
Affiliation:
Department of Mathematics and Statistics Simon Fraser UniversityBurnaby B.C. V5AIS6
Matatyahu Rubin
Affiliation:
Department of Mathematics Ben Gurion University of the Negev Beer Sheva, Israel
Charles Steinhorn
Affiliation:
Department of Mathematics Vassar College Poughkeepsie, New York USA 12601
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Abstract

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An ordered structure is o-minimal if every definable subset is the union of finitely many points and open intervals. A theory is o-minimal if all its models are ominimal. All theories considered will be o-minimal. A theory is said to be n-ary if every formula is equivalent to a Boolean combination of formulas in n free variables. (A 2-ary theory is called binary.) We prove that if a theory is not binary then it is not rc-ary for any n. We also characterize the binary theories which have a Dedekind complete model and those whose underlying set order is dense. In [5], it is shown that if T is a binary theory, is a Dedekind complete model of T, and I is an interval in , then for all cardinals K there is a Dedekind complete elementary extension of , so that . In contrast, we show that if T is not binary and is a Dedekind complete model of T, then there is an interval I in so that if is a Dedekind complete elementary extension of .

Keywords

Primary: 03C07secondary: 03C65
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 4 , 01 August 1992 , pp. 843 - 855
Copyright
Copyright © Canadian Mathematical Society 1992

References

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