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:

1.
$$

2.
$$
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 $\mathrm{(}\mathrm{Q}\mathrm{,}\mathrm{\le}\mathrm{)}$.
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  20130322 13:23:16 
Last modified on  20130322 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 