Let *M* be a structure in a language *L*. Let *L(M)* denote a new language, in which we have added a new constant symbol *a* for each element *a* of *M*. Then *M* can be expanded to an *L(M)*-structure in a tautological way.

The **diagram** of *M*, denoted diag(*M*) is the collection of all quantifier-free *L(M)*-statements true in *M*. The **elementary diagram** of *M*, denoted eldiag(*M*) is the collection of all *L(M)*-statements true in *M*.

The significance of these notions are that:

- Models of the diagram of
*M*are the same thing as*L*-structures extending*M*. - Models of the elementary diagram of
*M*are the same thing as elementary extensions of*M*. ::: ::: :::