For a structure A, the atomic diagram AtDiag(A) is the subset of the elementary diagram consisting only of the atomic sentences (no negations) in the language expanded by constants c_a naming each element of A that are true in A.

Principle

The atomic diagram records only positive atomic information about named elements: relations and equalities that hold. It is weaker than the elementary diagram because it omits explicit negations, yielding more flexible constraints on models that need only realize the asserted atoms.

Demonstration

With constants {c_a : a ∈ A}, collect all atomic sentences R(c_a1,...,c_an) and equalities among constants true in A. Any model of AtDiag(A) interprets the constants to satisfy those positive atomic facts; such models may still identify different constants unless equalities forbid it.

Misapplication

Assuming AtDiag(A) determines all atomic facts including non-holding ones; because it omits negated atoms, models of AtDiag(A) may satisfy additional atomic relations not present in A or identify distinct named elements, so one cannot reconstruct A uniquely from AtDiag(A) alone in general.

Consequence

Using AtDiag(A) suffices to embed A into models that realize its positive atomic structure and is often employed when one wants to freely allow additional relations or identifications while ensuring the named atoms hold; it facilitates partial amalgamation and directed constructions.

Reversal

The reversal is to include negations (recovering the full elementary diagram) or to consider only the complete theory without names. Adding negations restricts possible models and can force uniqueness of the named part, while removing names further abstracts the description.

Boundary

Applies in the expanded language with one constant per element and is limited to atomic positive assertions; it does not constrain non-atomic first-order consequences or forbid atoms absent from the list, so additional axioms are needed to control identifications and negative facts.

Semantic Tension

Tension with the elementary diagram: atomic diagrams are permissive and focused on positives, whereas elementary diagrams fix both presence and absence of atomic facts. Tension also exists between atomic diagrams and complete theories, which omit individual naming but capture global consequences.

Synthesis

The atomic diagram of A is the set of all positive atomic sentences in the language with constants naming A's elements that hold in A; it is a lighter-weight syntactic portrait that forces certain atoms while leaving negations and identifications open for further specification.