##  [Atomic Diagram](/atomic-diagram-0) 

 Definition

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

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

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

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

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

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

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

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

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.