In model theory, a branch of mathematical logic, the diagram of a structure is a simple but powerful concept for proving useful properties of a theory, for example the amalgamation property and the joint embedding property, among others.

Definition

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 a subset of the domain of ${\displaystyle {\mathfrak {A)).}$ Now one may expand ${\displaystyle {\mathfrak {A))}$ to the model

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

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 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 ${\displaystyle {\mathfrak {A))_{A}.}$^[1][2] Symbolically, ${\displaystyle D({\mathfrak {A)))=D^{+}({\mathfrak {A)))\cup \neg D^{-}({\mathfrak {A)))}$.