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

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

The diagram of ${\mathfrak {A}}$ is the set of all atomic sentences and negations of atomic sentences of ${\mathcal {L}}_{A}$ that hold in ${\mathfrak {U}}_{A}$ .[1][2]

gollark: This is VERY convoluted.

gollark: Ask them to send me their interpreter so I can observe it.

gollark: I wonder if helloboi has some kind of automated text scrambling program.

gollark: Why not have a button where you press it and it just randomly generates a schedule, and you click "submit" when it generates a good one?

gollark: It's useful as a *rough guide*.

References

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

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[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.