---
tag: S7MJVSP79N
kind: concept
name: Class operators H S P
msc: [08B99, 03C05]
sources:
  - url: https://en.wikipedia.org/wiki/Variety_(universal_algebra)
    locator: Birkhoff's variety theorem
---

## Summary

The class operators $H$, $S$ and $P$ act on classes of algebras of one signature.
They give the homomorphic images, the subalgebras and the direct products of the members of the class.
They are the three closure operations that appear in Birkhoff's characterisation of varieties.
Rules for composing them, such as $SH \le HS$ and $PS \le SP$, lead to Tarski's formula $V(K) = HSP(K)$ for the variety generated by a class $K$.

## Explanations

### L1 human

Start with some algebras, for example the single group $\Z$.
The operator $S$ adds all subgroups, such as $2\Z$.
The operator $H$ adds all homomorphic images, such as $\Z / 6\Z$.
The operator $P$ adds all direct products, such as $\Z \times \Z$.
Doing all three, in the order $P$, then $S$, then $H$, gives every abelian group.