Diagram (mathematical logic) explained
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,
D(akA)=D+(akA)\cup\negD-(akA)
.
See also
Notes and References
- Book: Hodges . Wilfrid . Model theory . registration . Wilfrid Hodges. 1993 . Cambridge University Press. 9780521304429.
- Book: Chang . C. C. . Keisler . H. Jerome . Chen Chung Chang. H. Jerome Keisler. Model Theory . 2012 . Dover Publications . 672 pages . Third.
