The diagram of is the set of atomic and negated atomic sentences true in the expansion by constants naming every element of . Its elementary diagram contains every sentence true in that expansion. Models of these diagrams correspond respectively to embeddings and elementary embeddings of .
The elementary diagram of is the set of all first-order sentences true in the expansion of by a constant symbol for each element. A model of the elementary diagram contains an elementary copy of .
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 becomes, after existentially closing its new names, a sentence in its existential theory. Joint satisfiability with therefore makes the full diagram finitely satisfiable with . Compactness gives the embedding, with inequalities ensuring injectivity and signed relation literals ensuring reflection. The reverse implication is preservation of existential truth under embeddings.
If satisfies the universal theory of , every existential sentence true in is true in , since its falsehood in would supply a universal negation. Apply the existential-theory embedding criterion to and the complete theory of to obtain . Conversely universal truth descends to substructures. The elementarily equivalent model need not be the original structure itself.
A first-order structure embeds into a model of exactly when . For the nontrivial direction apply the compactness theorem to . If a finite diagram conjunction were inconsistent with , its universal negation after replacing the new constants by variables would belong to and contradict its truth in .
If a type over a structure is finitely satisfiable in , then the elementary diagram of together with is finitely satisfiable. Compactness gives an elementary extension of realizing .
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In mathematical logic, a diagram is a graphical representation of relationships or structures that can help to visualize and analyze various logical concepts or proofs. Diagrams can take many forms, depending on the context in which they are used. One common type of diagram in logic is the Venn diagram, which illustrates set relationships and intersections, helping to visualize logical operations such as conjunction (AND), disjunction (OR), and negation (NOT).