Given a structure M and a language expanded by a constant for each element of M, the atomic diagram is the set of all atomic and negated atomic sentences in that expanded language that are true in M; it records the complete atomic information about M relative to those constants.

Principle

Encode a structure up to isomorphism over named elements by listing which atomic relations and equalities (and their negations) hold between the named constants; this transforms semantic information into a syntactic set of sentences.

Demonstration

If M has domain {a,b} and a binary relation R with R(a,b) true and R(b,a) false, then in the language with constants c_a,c_b the atomic diagram contains R(c_a,c_b) and ¬R(c_b,c_a), as well as sentences stating equalities c_a≠c_b and the atomic facts for function values.

Misapplication

Confusing the atomic diagram with the complete theory of M (which contains all first-order sentences true in M), or attempting to use an atomic diagram without naming all required elements, which yields incomplete atomic information.

Consequence

Atomic diagrams allow one to build or amalgamate structures by taking models of the diagram; they are the basic syntactic tool for encoding a particular structure with named elements in compactness and embedding arguments.

Reversal

Using only the full theory of M (without constants) loses the specific information about individual elements; conversely, listing higher-complexity sentences (beyond atomic/negated atomic) exceeds the atomic diagram's minimal descriptive role.

Boundary

The atomic diagram is tied to a particular choice of constants naming elements of M and contains only atomic or negated atomic sentences; it does not capture quantified or complex first-order information unless expanded by further sentences or schemas.

Semantic Tension

Tension exists between atomic, element-named descriptions (very concrete but local) and theory-level descriptions (general but abstract); the atomic diagram privileges element-level specificity over general theory-wide truths.

Synthesis

The atomic diagram is the syntactic list, in a language with constants for elements, of all atomic and negated atomic sentences true in a structure, providing a complete atomic-level blueprint for that structure relative to the chosen names.