Definable sets in Boolean ordered o-minimal structures. II

Journal of Symbolic Logic 68 (1):35-51 (2003)

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
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DOI 10.2178/jsl/1045861505
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Lattice Ordered O -Minimal Structures.Carlo Toffalori - 1998 - Notre Dame Journal of Formal Logic 39 (4):447-463.

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Weak Elimination of Imaginaries for Boolean Algebras.Roman Wencel - 2005 - Annals of Pure and Applied Logic 132 (2-3):247-270.
Imaginaries in Boolean Algebras.Roman Wencel - 2012 - Mathematical Logic Quarterly 58 (3):217-235.

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