Structure theorems in tame expansions of o-minimal structures by a dense set

The choice property in tame expansions of o‐minimal structures

“…Pairs with tame geometric behavior on the class of all definable sets have been extensively studied in the literature, and they include dense pairs [6], expansions of by a dense independent set [2], and expansions by a multiplicative group with the Mann property [8]. In [5], all these examples were unified under a common perspective, and a structure theorem was proved for their definable sets, in analogy with the cell decomposition theorem for o‐minimal structures. Namely, after imposing three conditions on the theory of [5, § 2, Assumptions (I)‐(III)], it was proved that every definable set is a finite union of cones .…”

“…In [5], all these examples were unified under a common perspective, and a structure theorem was proved for their definable sets, in analogy with the cell decomposition theorem for o‐minimal structures. Namely, after imposing three conditions on the theory of [5, § 2, Assumptions (I)‐(III)], it was proved that every definable set is a finite union of cones . We do not need to elaborate on the results from [5], but it is worth pointing out that they imply the failure of definable Skolem functions in that setting.…”

“…Namely, after imposing three conditions on the theory of [5, § 2, Assumptions (I)‐(III)], it was proved that every definable set is a finite union of cones . We do not need to elaborate on the results from [5], but it is worth pointing out that they imply the failure of definable Skolem functions in that setting. Fact Suppose that satisfies Assumptions (I)–(III) from [5].…”

“…We do not need to elaborate on the results from [5], but it is worth pointing out that they imply the failure of definable Skolem functions in that setting. Fact Suppose that satisfies Assumptions (I)–(III) from [5]. Let be definable.…”

A Theory of Pairs for Non-Valuational Structures

“…We let T P = Th(M P ) and assumeÑ = (N , N ) |= T P . (2) If Y ⊆ (N ) n is ∅-definable inÑ then it can be written as a boolean combination of sets defined by formulas of the form [3]) on expansions of o-minimal structures by dense predicates.…”