Boolean-Ordered O-Minimal Structures

Roman Wencel

The notions of o-minimality and quasi o-minimality for partially ordered structures were introduced by C. Toffalori in [To]. They are equivalent for Boolean ordered structures, as shown in [NW]. Moreover, in case of strongly o-minimal theory of a Boolean ordered structure, the prime model over an arbitrary set is unique up to isomorphism [NW]. There is a complete characterization of Boolean ordered structures with a small o-minimal theory [W].

We analyse properties of various kinds of atoms in Boolean algebras generated by algebraically closed subsets of o-minimal Boolean ordered structures and the way in which they relate to the notion of strong o-minimality. We also investigate the notion of orthogonality in Boolean ordered o-minimal structures and properties of algebraic closure.

[NW] L. Newelski and R. Wencel, Definable sets in Boolean-ordered o-minimal structures. I, JSL, submitted.

[To] C. Toffalori, Lattice ordered o-minimal structures, The Notre Dame Journal of Formal Logic, to appear.

[W] R. Wencel, Small theories of Boolean ordered o-minimal structures, in preparation.

Address: Wrocław University, Mathematical Institute, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

E-mail: Roman.Wencel@math.uni.wroc.pl