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 [math]\displaystyle{ \mathcal L }[/math] be a first-order language and [math]\displaystyle{ T }[/math] be a theory over [math]\displaystyle{ \mathcal L. }[/math] For a model [math]\displaystyle{ \mathfrak A }[/math] of [math]\displaystyle{ T }[/math] one expands [math]\displaystyle{ \mathcal L }[/math] to a new language

[math]\displaystyle{ \mathcal L_A := \mathcal L\cup \{c_a:a\in A\} }[/math]

by adding a new constant symbol [math]\displaystyle{ c_a }[/math] for each element [math]\displaystyle{ a }[/math] in [math]\displaystyle{ A, }[/math] where [math]\displaystyle{ A }[/math] is a subset of the domain of [math]\displaystyle{ \mathfrak A. }[/math] Now one may expand [math]\displaystyle{ \mathfrak A }[/math] to the model

[math]\displaystyle{ \mathfrak A_A := (\mathfrak A,a)_{a\in A}. }[/math]

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

The diagram [math]\displaystyle{ D(\mathfrak A) }[/math] of [math]\displaystyle{ \mathfrak A }[/math] is the set of all atomic sentences and negations of atomic sentences of [math]\displaystyle{ \mathcal L_A }[/math] that hold in [math]\displaystyle{ \mathfrak A_A. }[/math] Symbolically, [math]\displaystyle{ D(\mathfrak A) = D^+(\mathfrak A) \cup \neg D^-(\mathfrak A) }[/math].

See also

• Elementary diagram