---
tag: R2658B61VX
kind: definition
name: Prevariety
msc: ["08C15", "08B99"]
defines: EEAJ7AXBCT
uses: [PZMGQMDN25, WJ73XYBNE4]
sources:
  - url: https://en.wikipedia.org/wiki/Quasivariety
---

## Definition

Let $\sigma$ be an algebraic signature.
A class $K$ of $\sigma$-algebras is a *prevariety* if it satisfies these three conditions.

1. If $A \in K$ and $B$ is [[isomorphic|PZMGQMDN25]] to $A$, then $B \in K$.
2. If $A \in K$ and $B$ is a [[subalgebra|7CHG6SD365]] of $A$, then $B \in K$.
3. If $(A_i)_{i \in I}$ is a family of members of $K$, then the [[direct product|WJ73XYBNE4]] $\prod_{i \in I} A_i$ is in $K$; for $I = \emptyset$ this product is the one-element algebra.

## Remarks

In operator notation, $K$ is a prevariety exactly when $\mathbf{ISP}(K) = K$, and $\mathbf{ISP}(K)$ is the prevariety generated by $K$.

Example: residually finite groups form a prevariety.
A subgroup of a residually finite group is residually finite, by restricting the finite quotients.
A product of residually finite groups is residually finite.
Indeed, a nontrivial element has a nontrivial coordinate.
Compose the projection to that coordinate with a finite quotient that separates this coordinate from $1$.
This prevariety is not a [[quasivariety|WRXEKT4NQS]], since it is not closed under ultraproducts.
A nonprincipal ultraproduct of the groups $\Z / p\Z$ over the primes $p$ is a nontrivial torsion-free divisible abelian group, which has no proper subgroup of finite index.

Non-example: the class of fields fails condition 3, since $\Q \times \Q$ has the zero divisor $(1, 0)$.