Английская Википедия:Diagram (mathematical logic)

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Шаблон: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 <math>\mathcal L</math> be a first-order language and <math>T</math> be a theory over <math>\mathcal L.</math> For a model <math>\mathfrak A</math> of <math>T</math> one expands <math>\mathcal L</math> to a new language

<math>\mathcal L_A := \mathcal L\cup \{c_a:a\in A\}</math>

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

<math>\mathfrak A_A := (\mathfrak A,a)_{a\in A}.</math>

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

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

See also

• Elementary diagram

References

Шаблон:Mathlogic-stub