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
be a first-order language and
be a theory over
For a model
of
one expands
to a new language
![{\displaystyle {\mathcal {L))_{A}:={\mathcal {L))\cup \{c_{a}:a\in A\))](https://wikimedia.org/api/rest_v1/media/math/render/svg/2066c26d63fd41ee410202f8d95e54b458960ef6)
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
![{\displaystyle {\mathfrak {A))_{A}:=({\mathfrak {A)),a)_{a\in A}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e20c4676d68ab41bbc3c1b5f330c4a119567a89)
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,
.