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| Title of object: |
example of a universal structure |
| Canonical Name: |
ExampleOfUniversalStructure |
| Type: |
Example |
| Created on: |
2003-01-24 12:46:58 |
| Modified on: |
2005-01-19 10:21:30 |
| Classification: |
msc:03C50, msc:03C52 |
| Defines: |
dense, back and forth |
Revision comment (for changes between this and next version):
| it's not appropriate to define "dense" here, and it's now defined elsewhere |
Preamble:
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\def\Q{\mathbb{Q}} |
Content:
\PMlinkescapeword{proposition}
\PMlinkescapeword{satisfies}
Let $L$ be the first order language with the binary relation $\leq$. Consider the following sentences:
\begin{itemize}
\item $\forall x,y ((x\leq y \lor y\leq x) \land ((x \leq y \land y\leq x )\leftrightarrow x=y))$
\item $\forall x,y,z (x \leq y \land y \leq z \rightarrow x \leq z)$ \end{itemize}
Any $L$-structure satisfying these is called a linear order.
We define the relation $<$ so that $x<y$ iff $x \leq y \land x \neq y$. Now consider these sentences:
\begin{enumerate}
\item $\forall x,y ((x < y \rightarrow \exists z(x<z<y))$
\item $\forall x \exists y,z(y<x <z)$ \end{enumerate}
A linear order that satisfies 1. is called {\em dense}.
We say that a linear order that satisfies 2. is {\em without endpoints}.
Let $T$ be the theory of dense linear orders without endpoints. This is a complete theory.
\medskip
We can see that $(\Q,\leq)$ is a model of $T$. It is actually a rather special model.
\medskip
\begin{theorem}
\label{one} Let $(S,\leq)$ be any finite linear order. Then $S$ embeds in $(\Q,\leq)$.
\end{theorem}
Proof: By induction on $|S|$, it is trivial for $|S|=1$.
Suppose that the statement holds for all linear orders with cardinality less than or equal to $n$.
Let $|S|=n+1$, then pick some $a \in S$, let $S^{\prime}$ be the structure induced by $S$ on $S\setminus {a}$.
Then there is some embedding $e$ of $S^{\prime}$ into $\Q$.
\begin{itemize}
\item Now suppose $a$ is less than every member of $S^{\prime}$, then as $\Q$ is without endpoints, there is some element $b$ less than every element in the image of $e$.
Thus we can extend $e$ to map $a$ to $b$ which is an embedding of $S$ into $\Q$.
\item We work similarly if $a$ is greater than every element in $S^{\prime}$.
\item If neither of the above hold then we can pick some maximum $c_{1} \in S^{\prime}$ so that $c_{1}<a$.
Similarly we can pick some minimum $c_{2} \in S^{\prime}$ so that $c_{2}<a$. Now there is some $b \in \Q$ with $e(c_{1})<b<e(c_{2})$. Then extending $e$ by mapping $a$ to $b$ is the required embedding. $\square$ \end{itemize}
\medskip
It is easy to extend the above result to countable structures.
One views a countable structure as a the union of an increasing chain of finite substructures.
The necessary embedding is the union of the embeddings of the substructures. Thus $(\Q,\leq)$ is universal countable linear order.
\medskip
\begin{theorem} $(\Q,\leq)$ is homogeneous. \end{theorem}
Proof: The following type of proof is known as a {\em back and forth} argument.
Let $S_{1}$ and $S_{2}$ be two finite substructures of $(\Q,\leq)$.
Let $e:S_{1} \to S_{2}$ be an isomorphism.
It is easier to think of two disjoint copies $B$ and $C$ of $\Q$ with $S_{1}$ a substructure of $B$ and $S_{2}$ a substructure of $C$.
\medskip
Let $b_{1},b_{2},\ldots$ be an enumeration of $B \setminus S_{1}$.
Let $c_{1},c_{2},\ldots,c_{n}$ be an enumeration of $C \setminus S_{2}$. We iterate the following two step process:
{\bf The $i$th forth step} If $b_{i}$ is already in the domain of $e$ then do nothing. If $b_{i}$ is not in the domain of $e$.
Then as in proposition \ref{one}, either $b_{i}$ is less than every element in the domain of $e$ or greater than or it has an immediate successor and predecessor in the range of $e$.
Either way there is an element $c$ in $C \setminus$ range$(e)$ relative to the range of $e$.
Thus we can extend the isomorphism to include $b_{i}$.
{\bf The $i$th back step} If $c_{i}$ is already in the range of $e$ then do nothing. If $c_{i}$ is not in the domain of $e$.
Then exactly as above we can find some $b \in B \setminus $ dom$(e)$ and extend $e$ so that $e(b)=c_{i}$.
After $\omega$ stages, we have an isomorphism whose range includes every $b_{i}$ and whose domain includes every $c_{i}$. Thus we have an isomorphism from $B$ to $C$ extending $e$. $\square$
\medskip
A similar back and forth argument shows that any countable dense linear order without endpoints is isomorphic to $(\Q,\leq)$ so $T$ is $\aleph_{0}$-categorical. |
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