---
tag: Q94Z7TB8B5
kind: concept
name: Strong amalgamation property
msc: [08B25, 03C52]
specializes: [XSCSX39GEZ]
sources:
  - url: https://en.wikipedia.org/wiki/Amalgamation_property
---

## Summary

A class of algebras has the strong amalgamation property when any two members over a common part can be joined in a third member.
The join must identify only the common part.
It strengthens the [[amalgamation property|XSCSX39GEZ]], which allows the amalgam to identify more.
Groups and lattices have it, while fields have the amalgamation property but not the strong one.

## Explanations

### L1 human

Take two structures that share a common piece.
Amalgamation asks for a bigger structure containing both, with the two copies of the shared piece glued together.
Strong amalgamation asks in addition that nothing else gets glued: the two structures overlap exactly in the shared piece.
For finite graphs, gluing two graphs along a common subgraph and adding nothing else is such a strong amalgam.