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

${\mathcal {L))_{A}:={\mathcal {L))\cup \{c_{a}:a\in A\)$ by adding a new constant symbol $c_{a)$ for each element $a$ in $A,$ where $A$ is a subset of the domain of ${\mathfrak {A)).$ Now one may expand ${\mathfrak {A))$ to the model

${\mathfrak {A))_{A}:=({\mathfrak {A)),a)_{a\in A}.$ The positive diagram of ${\mathfrak {A))$ , sometimes denoted $D^{+}({\mathfrak {A)))$ , is the set of all those atomic sentences which hold in ${\mathfrak {A))$ while the negative diagram, denoted $D^{-}({\mathfrak {A))),$ thereof is the set of all those atomic sentences which do not hold in ${\mathfrak {A))$ .

The diagram $D({\mathfrak {A)))$ of ${\mathfrak {A))$ is the set of all atomic sentences and negations of atomic sentences of ${\mathcal {L))_{A)$ that hold in ${\mathfrak {A))_{A}.$ Symbolically, $D({\mathfrak {A)))=D^{+}({\mathfrak {A)))\cup \neg D^{-}({\mathfrak {A)))$ .