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

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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

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References

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  1. ^ Hodges, Wilfrid (1993). Model theory. Cambridge University Press. ISBN 9780521304429.
  2. ^ Chang, C. C.; Keisler, H. Jerome (2012). Model Theory (Third ed.). Dover Publications. pp. 672 pages.