Embedded o‐minimal structures

We prove the following two theorems on embedded o‐minimal structures: T heorem 1. Let ℳ ≺ be o‐minimal structures and let ℳ* be the expansion of ℳ by all traces in M of 1‐variable formulas in , that is all sets of the form φ( M , ā ) for ā ⊆ N and φ( x , ȳ ) ∈ ℒ(). Then , for any N‐formula ψ( x 1 , …, x k ), the set ψ( M k ) is ℳ* ‐definable . T heorem 2. Let be an ω 1 ‐saturated structure and let S be a sort in eq . Let be the ‐induced structure on S and assume that is o‐minimal . Then is stably embedded .