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

References

^ Hodges, Wilfrid (1993). Model theory. Cambridge University Press. ISBN 9780521304429.
^ Chang, C. C.; Keisler, H. Jerome (2012). Model Theory (Third ed.). Dover Publications. pp. 672 pages.