Definable sets in Boolean ordered o-minimal structures. II

Journal of Symbolic Logic 68 (1):35-51 (2003)
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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    Roman Wencel (2012). Imaginaries in Boolean Algebras. Mathematical Logic Quarterly 58 (3):217-235.
    Roman Wencel (2005). Weak Elimination of Imaginaries for Boolean Algebras. Annals of Pure and Applied Logic 132 (2-3):247-270.
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