## Journal of Symbolic Logic

### Definable sets in Boolean ordered o-minimal structures. II

Roman Wencel

#### Abstract

Let $(M,\leq,...)$ 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.

#### Article information

Source
J. Symbolic Logic, Volume 68, Issue 1 (2003), 35-51.

Dates
First available in Project Euclid: 21 February 2003

https://projecteuclid.org/euclid.jsl/1045861505

Digital Object Identifier
doi:10.2178/jsl/1045861505

Mathematical Reviews number (MathSciNet)
MR1959311

Zentralblatt MATH identifier
1043.03033

#### Citation

Wencel, Roman. Definable sets in Boolean ordered o-minimal structures. II. J. Symbolic Logic 68 (2003), no. 1, 35--51. doi:10.2178/jsl/1045861505. https://projecteuclid.org/euclid.jsl/1045861505

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