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

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

by adding a new constant symbol ${\displaystyle c_{a}}$ for each element ${\displaystyle a}$ in ${\displaystyle A,}$ where ${\displaystyle A}$ is the domain of ${\displaystyle {\mathfrak {A}}.}$ Now one may expand ${\displaystyle {\mathfrak {A}}}$ to the model

${\displaystyle {\mathfrak {A}}_{A}:=({\mathfrak {A}},a)_{a\in A}.}$

The diagram of ${\displaystyle {\mathfrak {A}}}$ is the set of all atomic sentences and negations of atomic sentences of ${\displaystyle {\mathcal {L}}_{A}}$ that hold in ${\displaystyle {\mathfrak {A}}_{A}.}$[1][2]

• Elementary diagram