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

Let $\mathcal L$ be a first-order language and $T$ be a theory over $\mathcal L.$ For a model $\mathfrak A$ of $T$ one expands $\mathcal L$ to a new language

: $\mathcal L_A := \mathcal L\cup \{c_a:a\in A\}$

by adding a new constant symbol $c_a$ for each element $a$ in $A,$ where $A$ is a subset of the domain of $\mathfrak A.$ Now one may expand $\mathfrak A$ to the model

: $\mathfrak A_A := (\mathfrak A,a)_{a\in A}.$

The positive diagram of $\mathfrak A$ , sometimes denoted $D^+(\mathfrak A)$ , is the set of all those atomic sentences which hold in $\mathfrak A$ while the negative diagram, denoted $D^-(\mathfrak A),$ thereof is the set of all those atomic sentences which do not hold in $\mathfrak A$ .

The diagram $D(\mathfrak A)$ of $\mathfrak A$ is the set of all atomic sentences and negations of atomic sentences of $\mathcal L_A$ that hold in $\mathfrak A_A.$ {{cite book|last1=Hodges |first1=Wilfrid |title=Model theory |url=https://archive.org/details/modeltheory0000hodg |url-access=registration |author-link=Wilfrid Hodges|date=1993 |publisher=Cambridge University Press|isbn=9780521304429}}{{cite book|last1=Chang |first1=C. C. |last2=Keisler |first2=H. Jerome |author-link1=Chen Chung Chang|author-link2=H. Jerome Keisler|title=Model Theory |date=2012 |publisher=Dover Publications |pages=672 pages |edition=Third}} Symbolically, $D(\mathfrak A) = D^+(\mathfrak A) \cup \neg D^-(\mathfrak A)$ .

References

Category:Mathematical logic

Category:Model theory

Diagram (mathematical logic)

Definition

See also

References