##  [Elementary Diagram](/elementary-diagram-1) 

 Definition

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

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

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

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

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

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

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

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

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.