example of a universal structure

Let $L$ be the first order language with the binary relation $\le$. Consider the following sentences:

• $\forall x,y((x\le y\vee y\le x)\wedge ((x\le y\wedge y\le x)\leftrightarrow x=y))$

• $\forall x,y,z(x\le y\wedge y\le z\to x\le z)$

Any $L$-structure satisfying these is called a linear order. We define the relation so that iff $x\le y\wedge x\ne y$. Now consider these sentences:

A linear order that satisfies 1. is called dense (http://planetmath.org/DenseTotalOrder). We say that a linear order that satisfies 2. is without endpoints. Let $T$ be the theory of dense linear orders without endpoints. This is a complete theory.

We can see that $(\mathbb{Q},\le )$ is a model of $T$. It is actually a rather special model.

Theorem 1

Let $\mathrm{(}S\mathrm{,}\mathrm{\le }\mathrm{)}$ be any finite linear order. Then $S$ embeds in $\mathrm{(}\mathrm{Q}\mathrm{,}\mathrm{\le }\mathrm{)}$.

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 $\mathbb{Q}$.

• Now suppose $a$ is less than every member of $S^{\prime }$, then as $\mathbb{Q}$ 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 $\mathbb{Q}$.

• We work similarly if $a$ is greater than every element in $S^{\prime }$.

• If neither of the above hold then we can pick some maximum $c_1\in S^{\prime }$ so that . Similarly we can pick some minimum $c_2\in S^{\prime }$ so that . Now there is some $b\in \mathbb{Q}$ with . Then extending $e$ by mapping $a$ to $b$ is the required embedding. $\mathrm{\square }$

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 $(\mathbb{Q},\le )$ is universal countable linear order.

Theorem 2

$(\mathbb{Q},\le )$ is homogeneous.

Proof: The following type of proof is known as a back and forth argument. Let $S_1$ and $S_2$ be two finite substructures of $(\mathbb{Q},\le )$. Let $e:S_1\to S_2$ be an isomorphism. It is easier to think of two disjoint copies $B$ and $C$ of $\mathbb{Q}$ with $S_1$ a substructure of $B$ and $S_2$ a substructure of $C$.

Let $b_1,b_2,\mathrm{\dots }$ be an enumeration of $B\setminus S_1$. Let $c_1,c_2,\mathrm{\dots },c_n$ be an enumeration of $C\setminus S_2$. We iterate the following two step process:

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 1, 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 $C\setminus \mathrm{range}(e)$ relative to the range of $e$. Thus we can extend the isomorphism to include $b_i$.

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 $b\in B\setminus \mathrm{dom}(e)$ 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 $e$. $\mathrm{\square }$

A similar back and forth argument shows that any countable dense linear order without endpoints is isomorphic to $(\mathbb{Q},\le )$ so $T$ is ${\mathrm{\aleph }}_0$-categorical.

Title	example of a universal structure
Canonical name	ExampleOfAUniversalStructure
Date of creation	2013-03-22 13:23:16
Last modified on	2013-03-22 13:23:16
Owner	uzeromay (4983)
Last modified by	uzeromay (4983)
Numerical id	15
Author	uzeromay (4983)
Entry type	Example
Classification	msc 03C50
Classification	msc 03C52
Related topic	Homogeneous4
Related topic	KappaCategorical
Related topic	DifferentialEquation
Related topic	ExampleOfDefinableType
Related topic	RandomGraph
Defines	back and forth