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

T h L (M) (M) = T h 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 $N$ .

Elementary substructure

See also: Elementary substructure, Formula, Model theory, N-tuple, Structures, Substructure, Tarski-Vaught test