##  [Elementary Diagram](/elementary-diagram-0) 

 Definition

For a structure M in a language L, the elementary diagram is the set of all first-order L(M)-sentences (where L(M) is L expanded with a constant for each element of M) that are true in the expanded structure; it records every first-order fact about named elements of M.

 

 

 

 

 

 





## Principle

Principle

Naming elements turns element-level truths into sentence-level assertions: by adjoining a constant for each element and including all sentences true in that expansion, the elementary diagram encodes the complete first-order theory of the particular named structure rather than just its theory up to isomorphism.

 

 

 

 

 





## Demonstration

Demonstration

Take the structure (N,+) and expand the language with constants c_n for each natural number n. The elementary diagram contains sentences like c_2 + c_3 = c_5, statements asserting particular equalities, inequalities, and any first-order property involving these constants that holds in (N,+).

 

 

 

 

## Misapplication

Misapplication

Equating the elementary diagram with the atomic diagram (which contains only atomic and negated-atomic formulas) or assuming the elementary diagram is a language-invariant object rather than relative to the chosen naming of elements.

 

 

 

 

 





## Consequence

Consequence

Knowing the elementary diagram of M determines, up to isomorphism respecting the constant names, all first-order consequences about the specific elements; it allows one to build embeddings and to phrase back-and-forth arguments concretely, and to reduce questions about M to satisfiability of sentences in the expanded language.

 

 

 

 

## Reversal

Reversal

Dropping the added constants yields the ordinary theory of M (the set of sentences true in M without names), which loses the identification of individual elements; conversely, two non-isomorphic structures can have the same theory but different elementary diagrams once elements are named.

 

 

 

 

 





## Boundary

Boundary

The elementary diagram is defined only within first-order syntax (or a chosen logic) and depends on the chosen expansion by constants; it does not by itself capture higher-order properties, cardinality-sensitive statements beyond first order, or information about undefinable sets not expressible with the available syntax.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension exists between 'elementary diagram' and 'atomic diagram' (the former includes all first-order sentences with constants, the latter only atomic truths), and between diagrams and complete theories: diagrams fix element names, theories forget them.

 

 

 

 

 





## Synthesis

Synthesis

The elementary diagram is the complete collection of first-order sentences in the language with constants naming every element of a structure that are true in that expansion; it concretely encodes the structure’s element-level first-order information and serves as a tool for constructing embeddings and transferring element-specific facts.