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]^[1]^[2] Symbolically, [math]\displaystyle{ D(\mathfrak A) = D^+(\mathfrak A) \cup \neg D^-(\mathfrak A) }[/math].

References

↑ Hodges, Wilfrid (1993). Model theory. Cambridge University Press. ISBN 9780521304429. https://archive.org/details/modeltheory0000hodg.
↑ Chang, C. C.; Keisler, H. Jerome (2012). Model Theory (Third ed.). Dover Publications. pp. 672 pages.

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[1] Hodges, Wilfrid (1993). Model theory. Cambridge University Press. ISBN 9780521304429. https://archive.org/details/modeltheory0000hodg.

[2] Chang, C. C.; Keisler, H. Jerome (2012). Model Theory (Third ed.). Dover Publications. pp. 672 pages.

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Diagram (Mathematical Logic)

Definition

See also

References