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Embedded o‐minimal structures
Author(s) -
Hasson Assaf,
Onshuus Alf
Publication year - 2010
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdp098
Subject(s) - mathematics , sort , structured program theorem , combinatorics , set (abstract data type) , discrete mathematics , arithmetic , computer science , programming language
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 .