---
tag: XSCSX39GEZ
kind: concept
name: Amalgamation property
msc: [03C52]
sources:
  - url: https://en.wikipedia.org/wiki/Amalgamation_property
same_as:
  wikidata: Q4739384
  wikipedia: Amalgamation property
---

## Summary

A class of structures has the amalgamation property when any two members that share a common part fit together in a third member.
In the third member, the two copies of the common part must coincide.
It is one of the three conditions in Fraïssé's theorem, which builds a countable homogeneous structure, the [[Fraïssé limit|HJVRBDQSGG]], from a class of finite structures.
It also appears in algebra, for example in the study of amalgamated free products of groups.

## Explanations

### L1 human

Think of two drawings that both contain the same small picture.
Amalgamation means one can draw a bigger picture that contains both drawings and in which the two copies of the small picture are the same.
For finite graphs this is easy: take two graphs $B$ and $C$ that both contain a graph $A$, glue them along $A$, and add no other edges.
The glued graph contains $B$ and $C$, and the two copies of $A$ coincide.