Elementary substructure
From Exampleproblems
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 
, for every formula 
of the language L, we have that 
if and only if 
.
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 is an elementary substructure of N.
