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