Diagram (mathematical logic)

Let ${\displaystyle {\mathcal {L}}}$ be a first-order language and ${\displaystyle T}$ be a theory over ${\displaystyle {\mathcal {L}}.}$ For a model ${\displaystyle {\mathfrak {A}}}$ of ${\displaystyle T}$ one expands ${\displaystyle {\mathcal {L}}}$ to a new language

${\displaystyle {\mathcal {L}}_{A}:={\mathcal {L}}\cup \{c_{i}:a_{i}\in A\}}$

by adding a new constant symbol ${\displaystyle c_{i}}$ for each element ${\displaystyle a_{i}}$ in ${\displaystyle A,}$ where ${\displaystyle A}$ is a (not necessarily proper) subset of the domain of ${\displaystyle {\mathfrak {A}}.}$ Now one may expand ${\displaystyle {\mathfrak {A}}}$ to the model ${\displaystyle {\mathfrak {A}}_{A}:=({\mathfrak {A}},a_{i})_{a_{i}\in A}.}$

The (atomic) diagram ${\displaystyle D({\mathfrak {A}})}$ of ${\displaystyle {\mathfrak {A}}}$ is the set of all atomic sentences and negations of atomic sentences of ${\displaystyle {\mathcal {L}}_{A}}$ that hold in ${\displaystyle {\mathfrak {A}}_{A}.}$^[2][3] Symbolically, ${\displaystyle D({\mathfrak {A}})=\{\phi$ :{\mathfrak {A}}\models \phi {\text{ where }}\phi {\text{ is an atomic }}{\mathcal {L}}_{A}{\text{-formula or a negation thereof}}\}} .

The elementary diagram ${\displaystyle D_{el}({\mathfrak {A}})}$ of ${\displaystyle {\mathfrak {A}}}$ is the set of all sentences of ${\displaystyle {\mathcal {L}}_{A}}$ that hold in ${\displaystyle {\mathfrak {A}}_{A}.}$^[4][5] Symbolically, ${\displaystyle D_{el}({\mathfrak {A}})=\{\phi$ :{\mathfrak {A}}\models \phi {\text{ where }}\phi {\text{ is a sentence of }}{\mathcal {L}}_{A}\}} . Equivalently, ${\displaystyle D_{el}({\mathfrak {A}})=\{\phi (c_{1},\ldots ,c_{n}):{\mathfrak {A}}\models \phi (c_{1},\ldots ,c_{n}){\text{ where }}\phi {\text{ is an }}{\mathcal {L}}{\text{-formula}}\}}$.^[6]

The positive diagram of ${\displaystyle {\mathfrak {A}}}$, sometimes denoted ${\displaystyle D^{+}({\mathfrak {A}})}$, is the set of all those atomic sentences which hold in ${\displaystyle {\mathfrak {A}}}$ while the negative diagram, denoted ${\displaystyle D^{-}({\mathfrak {A}}),}$ thereof is the set of all those atomic sentences which do not hold in ${\displaystyle {\mathfrak {A}}}$. The positive diagram and the negation of all of the formulas in the negative diagram combine to form the atomic diagram. Symbolically, ${\displaystyle D({\mathfrak {A}})=D^{+}({\mathfrak {A}})\cup \neg D^{-}({\mathfrak {A}})}$.