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

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

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

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

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

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

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

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

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.