---
tag: HJVRBDQSGG
kind: concept
name: Fraïssé limit
msc: [03C15, 03C52]
sources:
  - url: https://en.wikipedia.org/wiki/Fraïssé_limit
same_as:
  wikidata: Q85850626
  wikipedia: Fraïssé limit
---

## Summary

A Fraïssé limit of a class of finitely generated structures is a countable structure that is as symmetric as possible.
Its finitely generated substructures are exactly the members of the class, up to isomorphism.
Here "as symmetric as possible" means ultrahomogeneous: every isomorphism between finitely generated substructures extends to an automorphism.
Fraïssé's theorem gives existence and uniqueness up to isomorphism under four conditions.
The class must have the hereditary, joint embedding and [[amalgamation|XSCSX39GEZ]] properties, and only countably many isomorphism types.
The rational numbers with their order and the [[Rado graph|RMXR0FEF54]] are the standard examples.

## Explanations

### L1 human

Start with a collection of small pieces, such as all finite linear orders or all finite graphs.
The Fraïssé limit is one countable object into which every piece fits, and in which any two copies of the same piece look alike from the outside.
For finite linear orders the limit is $(\Q, <)$.
Every finite order sits inside $\Q$, and any order-preserving map between two finite subsets of $\Q$ extends to an order-preserving bijection of all of $\Q$.