Diagram (mathematical logic)

Template:Short description 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 $ℒ$ be a first-order language and $T$ be a theory over $ℒ.$ For a model $𝔄$ of $T$ one expands $ℒ$ to a new language

${ℒ}_A:=ℒ\cup \{c_a:a\in A\}$

by adding a new constant symbol $c_a$ for each element $a$ in $A,$ where $A$ is a subset of the domain of $𝔄.$ Now one may expand $𝔄$ to the model

${𝔄}_A:=(𝔄,a{)}_{a\in A}.$

The positive diagram of $𝔄$, sometimes denoted $D^{+}(𝔄)$, is the set of all those atomic sentences which hold in $𝔄$ while the negative diagram, denoted $D^{-}(𝔄),$ thereof is the set of all those atomic sentences which do not hold in $𝔄$.

The diagram $D(𝔄)$ of $𝔄$ is the set of all atomic sentences and negations of atomic sentences of ${ℒ}_A$ that hold in ${𝔄}_A.$ Symbolically, $D(𝔄)=D^{+}(𝔄)\cup \mathrm{\neg }{D}^{-}(𝔄)$.

See also

• Elementary diagram

References

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