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Definable sets in boolean ordered o-minimal structures. II

Published online by Cambridge University Press:  12 March 2014

Roman Wencel
Roman Wencel*
Affiliation:
Mathematical Institute, University of Wrocław, PL. Grunwaldzki 2/4, 50-384 Wrocław, Poland, E-mail: [email protected]
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Abstract

Let (M, ≤,…) denote a Boolean ordered o-minimal structure. We prove that a Boolean subalgebra of M determined by an algebraically closed subset contains no dense atoms. We show that Boolean algebras with finitely many atoms do not admit proper expansions with o-minimal theory. The proof involves decomposition of any definable set into finitely many pairwise disjoint cells, i.e., definable sets of an especially simple nature. This leads to the conclusion that Boolean ordered structures with o-minimal theories are essentially bidefinable with Boolean algebras with finitely many atoms, expanded by naming constants. We also discuss the problem of existence of proper o-minimal expansions of Boolean algebras.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 1 , March 2003 , pp. 35 - 51
Copyright
Copyright © Association for Symbolic Logic 2003

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References

REFERENCES

[KPS]Knight, J. F., 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
[LS]Laskowski, C. and Steinhorn, C., On o-minimal expansions of archimedean ordered groups, this Journal, vol. 60 (1995), pp. 817–831.Google Scholar
[NW]Newelski, L. and Wencel, R., Definable sets in Boolean ordered o-minimal structures. I, this Journal, vol. 66 (2001), pp. 1821–1836.Google 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
[To]Toffalori, C., Lattice ordered o-minimal structures, Notre Dame Journal of Formal Logic, vol. 39 (1998), pp. 447–463.CrossRefGoogle Scholar
[We]Wencel, R., Small theories of Boolean ordered o-minimal structures, this Journal, vol. 67 (2001), no. 4, pp. 1385–1390.Google Scholar