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Preprint Number 1958

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1958. Masato Fujita
Almost o-minimal structures and X-structures

Submission date: 3 April 2021


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 {A_λ}_{λ in Λ} be a finite family of definable subsets of M^{m+n}. Take an arbitrary positive element R in M and set B=]-R,R[^n. Then, there exists a finite partition into definable sets

M^m × B = X_1 ∪ ... ∪ X_k
such that B=(X_1)_b ∪ ... ∪ (X_k)_b is a definable cell decomposition of B for any b in M^m and either X_i ∩ A_λ = ∅ or X_i ⊆ A_λ for any 1 ≤ i ≤ k and λ in Λ. Here, the notation S_b denotes the fiber of a definable subset S of M^{m+n} at b in M^m. 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.

Mathematics Subject Classification: Primary 03C64, Secondary 14P99

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Full text arXiv 2104.01312: pdf, ps.

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