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 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 ${\displaystyle {\mathfrak {U}}_{A}}$.[1][2]

References

1. Hodges, Wilfrid (1993). Model theory. Cambridge University Press.
2. Chang, C. C.; Keisler, H. Jerome (2012). Model Theory (Third ed.). Dover Publications. pp. 672 pages.