Fraïssé limit

Remarks

Fraïssé's theorem: suppose $\mathcal{K}$ is nonempty and has countably many isomorphism types. Suppose it has the hereditary property, the joint embedding property and the amalgamation property. Then $\mathcal{K}$ has a Fraïssé limit, and any two Fraïssé limits of $\mathcal{K}$ are isomorphic. Conversely, the age of a countable ultrahomogeneous structure has these properties.

Example: $(\Q, <)$ is the Fraïssé limit of the class of finite linear orders. Every finite linear order embeds in $\Q$, so clause 2 holds. For clause 3, an order isomorphism between finite subsets of $\Q$ extends piecewise linearly to an order automorphism of $\Q$.

Non-example: $(\Z, <)$ is countable and its age is again the class of finite linear orders, but it fails clause 3. The map $0 \mapsto 0$, $1 \mapsto 2$ is an isomorphism between the substructures $\set{0, 1}$ and $\set{0, 2}$. It has no extension to an automorphism, because no integer lies strictly between $0$ and $1$, while $1$ lies strictly between $0$ and $2$.