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: What I don't like is when people go from "hmm yes I dislike this" to "this person clearly must be prevented from sharing opinions anywhere".

gollark: See, that's unreasonable, getting angry at people is fine.

gollark: I will probably, to some amount of personal cost, defend people's right to say things I dislike, but that doesn't mean I have to agree with them or particularly support said things.

gollark: You are not going to make people budge on their opinions by saying "no, this opinion is illegal now" or something.

gollark: Okay, too bad, don't let them do much based on it I guess.

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.