elementary embedding

Let τ be a signaturePlanetmathPlanetmathPlanetmath and 𝒜 and be two structuresMathworldPlanetmath for τ such that f:𝒜 is an embeddingPlanetmathPlanetmathPlanetmath. Then f is said to be elementary if for every first-order formulaMathworldPlanetmathPlanetmath ϕF(τ), we have


In the expression above, 𝒜ϕ means: if we write ϕ=ϕ(x1,,xn) where the free variablesMathworldPlanetmathPlanetmath of ϕ are all in {x1,,xn}, then ϕ(a1,,an) holds in 𝒜 for any ai𝒜 (the underlying universePlanetmathPlanetmath of 𝒜).

If 𝒜 is a substructure of such that the inclusion homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath is an elementary embedding, then we say that 𝒜 is an elementary substructure of , or that is an elementary extension of 𝒜.

Remark. A chain 𝒜1𝒜2𝒜n of τ-structures is called an elementary chain if 𝒜i is an elementary substructure of 𝒜i+1 for each i=1,2,. It can be shown (Tarski and Vaught) that


is a τ-structure that is an elementary extension of 𝒜i for every i.

Title elementary embedding
Canonical name ElementaryEmbedding
Date of creation 2013-03-22 13:00:29
Last modified on 2013-03-22 13:00:29
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Definition
Classification msc 03C99
Synonym elementary monomorphism
Defines elementary substructure
Defines elementary extension
Defines elementary chain