Diagram (mathematical logic)

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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 be a

first-order language
and be a theory over For a
model
of one expands to a new language

by adding a new constant symbol for each element in where is a subset of the domain of Now one may expand to the model

The positive diagram of , sometimes denoted , is the set of all those atomic sentences which hold in while the negative diagram, denoted thereof is the set of all those atomic sentences which do not hold in .

The diagram of is the set of all atomic sentences and negations of atomic sentences of that hold in [1][2] Symbolically, .

See also

References