Given a structure A, the elementary diagram Diag(A) is the set of all atomic and negated atomic sentences in the language expanded by naming each element a ∈ A with a new constant symbol c_a, that are true in A. It records complete atomic-level information about A with constants.

Principle

The elementary diagram encodes the structure's full atomic and negated atomic information once elements are named; it is a convenient theory in an expanded language whose models are precisely the expansions that realize A's atomic truths about named elements.

Demonstration

For structure A, introduce constant symbols {c_a : a ∈ A} and collect all sentences of the form R(c_a1,...,c_an) or ¬R(c_a1,...,c_an) and equalities/inequalities among constants that hold in A. Any model of Diag(A) in the expanded language contains an isomorphic copy of A interpreting c_a as the corresponding element.

Misapplication

Confusing the elementary diagram with the complete theory Th(A): Diag(A) is sensitive to names for individual elements and is much stronger (often inconsistent with other intended identifications), while Th(A) is a set of sentences without element names and is invariant under isomorphism.

Consequence

Diag(A) lets one force the presence of a particular substructure when building models: a model of Diag(A) necessarily interprets the added constants to satisfy the same atomic facts as A, so diagrams are used in amalgamation, realizability, and constructing elementary extensions or embeddings.

Reversal

The dual notion is the omission of the diagram: considering only the theory without constants or only the atomic diagram (see next entry). Omitting negations or constants weakens the forcing power and may allow models that do not include a copy of A.

Boundary

Defined in the language expanded by distinct constant symbols naming each element; it applies at the atomic/negated-atomic level and does not include arbitrary first-order consequences unless they are logically implied by the atomic set; careful control of the expanded signature is required.

Semantic Tension

Competes with other syntactic encodings like canonical diagrams, atomic diagrams, and complete theories: the elementary diagram is fine-grained and parameterized by names, whereas general theories abstract away from particular elements and focus on sentence-level properties.

Synthesis

The elementary diagram of A is the theory in the language with constant names for each element listing every atomic and negated atomic sentence true in A; it is a precise syntactic portrait that forces models to interpret those names as a structure isomorphic on the named part to A.