Diagrams - Model Theory - Cambridge III notes

10 Diagrams

Let $N, M$ be $L$ -structures.

Remark 10.1. If $h : M \to N$ is an (elementary) $L$ -embedding then after identifying $a \in M$ with $h (a) \in N$ we can view $M$ as an (elementary) substructure of $N$ .

Given $A \subseteq M$ , let $L_{A} = L \cup {\underset{̲}{a} : a \in A}$ where $\underset{̲}{a}$ is a new constant symbol. Then $M$ is an $L_{A}$ -structure. Interpret $\underset{̲}{a}$ as $a$ .

Definition 10.2 (Diagram). The diagram of $M$ (respectively elementary diagram), $D (M)$ , is the set of quantifier-free $L_{M}$ -sentences (respectively all $L_{M}$ -sentences) true in $M$ .

Proposition 10.3. Assuming that:

$M$ is an $L$ -structure
$N^{*}$ an $L_{M}$ -structure such that $N^{*} ⊨ D (M)$
let $N$ be the $L$ -reduct of $N^{*}$ to $L$ (means throw away $L_{M} ∖ L$ sentences)
define $h : M \to N$ such that $h (a) = \underset{̲}{a} = a^{N}$ .

Then

h

is an

L

-embedding. Moreover, if

N^{*} ⊨ {T h}_{M} (M)

then

h

is an elementary

L

-embedding.

Proof. We use Corollary 2.4. Let $φ (x_{1}, \dots, x_{n})$ be a quantifier-free $L$ -formula, and fix $a_{1}, \dots, a_{n} \in M$ . Then

\begin{array}{l} M ⊨ φ (a_{1}, \dots, a_{n}) & ⟺ φ (\underset{̲}{a_{1}}, \dots, \underset{̲}{a_{n}}) \in D (M) \\ ⟺ N^{*} ⊨ φ (\underset{̲}{a_{1}}, \dots, \underset{̲}{a_{n}}) \\ ⟺ N ⊨ φ (h (a_{1}), \dots, h (a_{n})) \end{array}

Therefore $h$ is an $L$ -embedding.

“Moreover” is similar. □

Remark. You can use this to show that any torsion free abelian group is orderable ( Example Sheet 2).