---
tag: EEAJ7AXBCT
kind: concept
name: Prevariety
msc: ["08C15", "08B99"]
sources:
  - url: https://en.wikipedia.org/wiki/Quasivariety
---

## Summary

A prevariety is a class of algebras of one signature that is closed under isomorphic copies, [[subalgebras|7CHG6SD365]] and [[direct products|WJ73XYBNE4]].
Every [[quasivariety|WRXEKT4NQS]] and every [[variety|AJNZJD8442]] is a prevariety, but not conversely.
The prevariety generated by a class $K$ consists of the algebras that embed into a product of members of $K$.
Every prevariety is reflective: each algebra has a best approximation inside the class, as in the [[reflection onto a variety|6EWEMZ6FX1]].

## Explanations

### L1 human

Think of a class of algebras that you can only enlarge by two moves: taking a part (a subalgebra) and putting several members side by side (a product).
If the class is stable under both moves, it is a prevariety.
For example, residually finite groups form a prevariety: a subgroup or a product of residually finite groups is again residually finite.
Fields do not, because the product of two fields has zero divisors.