---
tag: M4ZHB3GXBP
kind: statement
role: theorem
name: Tarski's HSP formula
msc: [08B99, 03C05]
uses: [AJNZJD8442, S7MJVSP79N]
about: [AJNZJD8442, S7MJVSP79N]
sources:
  - url: https://en.wikipedia.org/wiki/Variety_(universal_algebra)
---

## Statement

Let $K$ be a class of algebras of a fixed signature.
Let $V(K)$ be the smallest class that contains $K$ and is closed under the [[class operators|S7MJVSP79N]] $H$, $S$ and $P$.
Then
$$V(K) = HSP(K).$$
By Birkhoff's HSP theorem, $HSP(K)$ is then the [[variety|AJNZJD8442]] generated by $K$.

## Remarks

Each of $H$, $S$ and $P$ maps $V(K)$ into itself, so $HSP(K) \subseteq V(K)$.
It remains to show that $HSP(K)$ is closed under $H$, $S$ and $P$.
This follows from the rules $SH \le HS$, $PS \le SP$, $PH \le HP$ and from the idempotence of each operator.
For example, $S\,HSP \le HS\,SP = HSP$ and $P\,HSP \le HP\,SP \le HSP\,P = HSP$.
Example: for $K = \set{\Z}$ in the signature of groups, $HSP(K)$ is the class of all abelian groups.
Indeed, every abelian group is a quotient of a free abelian group, and a free abelian group embeds in a power of $\Z$.

## Explanations

### L1 human

To build the smallest variety containing some given algebras, it is enough to apply three steps once each.
First take products, then subalgebras of those products, then homomorphic images.
No further rounds of the three constructions produce anything new.

### L2 human

The formula gives the standard way to find the members of $V(A)$ for one algebra $A$.
Each member of $V(A)$ is a homomorphic image of a subalgebra of a power $A^I$.
In particular, the free algebras of $V(A)$ are subalgebras of powers of $A$, which is the starting point of the study of free spectra.
In the congruence-distributive case, Jónsson's lemma sharpens the formula for the subdirectly irreducible members.