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 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 $$\mathcal L_A$$ that hold in $$\mathfrak A_A.$$ Symbolically, $$ D(\mathfrak A) = D^+(\mathfrak A) \cup \neg D^-(\mathfrak A)$$.