Diagram (mathematical logic)

Short description: Concept in model theory

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 $\mathfrak{𝔄}$ 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 $\mathfrak{𝔄}.$ 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 ${{ℒ}}_A$ that hold in ${\mathfrak{𝔄}}_A.$ Symbolically, $D(\mathfrak{𝔄})=D^{+}(\mathfrak{𝔄})\cup \mathrm{\neg }{D}^{-}(\mathfrak{𝔄})$.

See also

• Elementary diagram