Prevariety

Builds on Isomorphism Direct product of algebras

Definition

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

If $A \in K$ and $B$ is isomorphic to $A$ , then $B \in K$ .
If $A \in K$ and $B$ is a subalgebra of $A$ , then $B \in K$ .
If $(A_i)_{i \in I}$ is a family of members of $K$ , then the direct product $\prod_{i \in I} A_i$ 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, 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 $(1, 0)$ .

Reading path

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Tag	`R2658B61VX`
URI	`m2://R2658B61VX`
Kind	definition
Writing	W0 stub
MSC	08C15, 08B99
Defines	Prevariety
Uses	Isomorphism, Direct product of algebras
Changed	2 November 2026

Prevariety

Definition

Remarks

Reading path

Further reading

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