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.