Method of diagrams

The method of diagrams adds the diagram or elementary diagram of a structure to a theory and applies compactness. A resulting model supplies an embedding or elementary embedding of the named structure.

A finite conjunction from the signed atomic diagram of $A$ becomes, after existentially closing its new names, a sentence in its existential theory. Joint satisfiability with $T$ therefore makes the full diagram finitely satisfiable with $T$. Compactness gives the embedding, with inequalities ensuring injectivity and signed atomic formulas ensuring both preservation and reflection of relations as required for a structure embedding. The reverse implication is preservation of existential truth under embeddings.

If $B$ satisfies the universal theory of $A$, every existential sentence true in $B$ is true in $A$, since its falsehood in $A$ would supply a universal negation. Apply the existential-theory embedding criterion to $B$ and the complete theory of $A$ to obtain $A^{\ast }$. Conversely universal truth descends to substructures. The elementarily equivalent model need not be the original structure itself.

A first-order structure $A$ embeds into a model of $T$ exactly when $A\models T_{\forall }$. For the nontrivial direction apply the compactness theorem to $T\cup \mathrm{Diag}(A)$. If a finite diagram conjunction were inconsistent with $T$, its universal negation after replacing the new constants by variables would belong to $T_{\forall }$ and contradict its truth in $A$.

If a type $p(x)$ over a structure $M$ is finitely satisfiable in $M$, then the elementary diagram of $M$ together with $p(c)$ is finitely satisfiable. Compactness gives an elementary extension of $M$ realizing $p$.