---
tag: WRXEKT4NQS
kind: concept
name: Quasivariety
msc: ["08C15", "03C52"]
specializes: [EEAJ7AXBCT, 9A4CYWXPEA]
sources:
  - url: https://en.wikipedia.org/wiki/Quasivariety
same_as:
  wikidata: Q2752716
  wikipedia: Quasivariety
---

## Summary

A quasivariety is a class of algebras of one signature that is axiomatized by [[quasi-identities|94GVY3APPQ]].
Every [[variety|AJNZJD8442]] is a quasivariety, but quasivarieties need not be closed under homomorphic images.
By [[Mal'cev's characterization|DFNM3BGMAY]], they are the classes closed under isomorphic copies, subalgebras, direct products and ultraproducts.
They are the algebraic counterpart of universal Horn logic and appear in algebraic logic, where many logics are described by quasivarieties rather than varieties.

## Explanations

### L1 human

A variety is described by equations that must always hold.
A quasivariety is described by conditional equations: "if these equations hold, then that one holds too".
Torsion-free groups form a quasivariety: for each $n \ge 1$, the rule "if $x^n = 1$ then $x = 1$" is a quasi-identity.
They do not form a variety, because the torsion-free group $\Z$ has the quotient $\Z / 2\Z$, which has torsion.