---
tag: HHNWHS1XTN
kind: concept
name: Free product of algebras
msc: [08B25, "20E06"]
sources:
  - url: https://en.wikipedia.org/wiki/Free_product
---

## Summary

The free product of a family of algebras in a [[variety|AJNZJD8442]] is the most general algebra of the variety generated by images of the members.
It is the coproduct in the category of algebras of the variety.
The [[free product of groups|TM7QS7P7X9]] and the free product of monoids are the best known cases.

## Explanations

### L1 human

To form a free product, put the algebras together and let their elements combine with each other only as the laws of the variety demand.
No other relation between elements of different factors is added.
In the variety of groups, the free product of two copies of $\Z$ is the free group on two generators.
In the variety of abelian groups, the free product of $\Z$ and $\Z$ is $\Z \times \Z$, because the laws of the variety force the two generators to commute.