---
tag: TH2XQGGFWH
kind: concept
name: Residually small variety
msc: [08B26]
sources:
  - url: https://en.wikipedia.org/wiki/Residual_property_(mathematics)
---

## Summary

A [[variety|AJNZJD8442]] is residually small when the sizes of its [[subdirectly irreducible|DASF939439]] members are bounded by a cardinal.
By Birkhoff's subdirect representation theorem, every algebra of a variety is a subdirect product of subdirectly irreducible members, so these members control the whole variety.
A residually small variety has, up to isomorphism, only a set of subdirectly irreducible members; otherwise it is called residually large.
Residual smallness is related to the existence of enough injective algebras in a variety.

## Explanations

### L1 human

Every algebra of a variety can be broken down into irreducible pieces that cannot be split further.
In a residually small variety these pieces never get too big: there is a fixed size that none of them reaches.
Abelian groups form such a variety: their irreducible pieces are the cyclic groups $\Z/p^n\Z$ and the Prüfer groups $\Z(p^\infty)$, all countable.
The variety of all groups is not residually small, because there are simple groups of every infinite size.