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# 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 < N) 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

ThL(M)(M) = ThL(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.

03-10-2013 05:06:04