Diagram (mathematical logic)

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, ${\displaystyle D({\mathfrak {A}})=D^{+}({\mathfrak {A}})\cup \neg D^{-}({\mathfrak {A}})}$.

See also

• Elementary diagram