Submission date: 12 December 2019

Abstract:

We demonstrate the following uniform local definable cell decomposition theorem in this paper.

Consider a structure M = (M, <,0,+, ...) elementarily equivalent to a locally o-minimal expansion of the group of reals ( ℝ, <,0,+). Let {A_λ}_{λ in Λ} be a finite family of definable subsets of M^{m+n}. There exist an open box B in M^n containing the origin and a finite partition of 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 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.

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Full text arXiv 1912.05782: pdf, ps.