Almost o-minimal structures and [formula omitted]-structures

Abstract

We propose new structures called almost o-minimal structures and X-structures. The former is a first-order expansion of a dense linear order without endpoints such that the intersection of a definable set with a bounded open interval is a finite union of points and open intervals. The latter is a variant of van den Dries and Miller's analytic geometric categories and Shiota's X-sets and Y-sets. In them, the family of definable sets are closed only under proper projections unlike first-order structures. We demonstrate that an X-expansion of an ordered divisible abelian group always contains an o-minimal expansion of an ordered group such that all bounded X-definable sets are definable in the structure.Another contribution of this paper is a uniform local definable cell decomposition theorem for almost o-minimal expansions of ordered groups M=(M,<,0,+,…). Let {Aλ}λ∈Λ be a finite family of definable subsets of Mm+n, where m and n are positive integers. Take an arbitrary positive element R∈M and set B=]−R,R[n. Then, there exists a finite partition into definable setsMm×B=X1∪…∪Xk such that B=(X1)b∪…∪(Xk)b is a definable cell decomposition of B for any b∈Mm and either Xi∩Aλ=∅ or Xi⊆Aλ for any 1≤i≤k and λ∈Λ. Here, the notation Sb denotes the fiber of a definable subset S of Mm+n at b∈Mm. We introduce the notion of multi-cells and demonstrate that any definable set is a finite union of multi-cells in the course of the proof of the above theorem.