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

See also

• Elementary diagram

References

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