Elementary Substructure

In model theory, given two structures M and N in the same language L, we say that M is an elementary substructure of N (notated sometimes M) if 1. M is a substructure of N, and 2. for every finite tuple a\in M, for every formula \varphi(x) of the language L, we have that M\models \varphi(a) if and only if N\models \varphi(a). The second part may also be presented as saying that
Th_{L(M)}(M) = Th_{L(M)}(N).
The Tarski-Vaught test is very useful in determining whether, given a pair M\subset N, M is an elementary substructure of N.

 

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