---
tag: J037NW1Q38
kind: definition
name: Class operators H S P
msc: [08B99, 03C05]
defines: S7MJVSP79N
uses: [KBXSZC6BV1, PZMGQMDN25, WJ73XYBNE4]
sources:
  - url: https://en.wikipedia.org/wiki/Variety_(universal_algebra)
    locator: Birkhoff's variety theorem
---

## Definition

Let $K$ be a class of algebras of a fixed signature $\sigma$.

1. $I(K)$ is the class of algebras [[isomorphic|PZMGQMDN25]] to a member of $K$.
2. $H(K)$ is the class of algebras $B$ such that there is a surjective [[homomorphism|KBXSZC6BV1]] $A \to B$ for some $A \in K$.
3. $S(K)$ is the class of algebras isomorphic to a [[subalgebra|7CHG6SD365]] of a member of $K$.
4. $P(K)$ is the class of algebras isomorphic to a [[direct product|WJ73XYBNE4]] $\prod_{i \in I} A_i$ with all $A_i \in K$ and $I$ any index set.

For operators $O_1, O_2$, the composite $O_1 O_2$ is defined by $O_1 O_2(K) = O_1(O_2(K))$.
One writes $O_1 \le O_2$ when $O_1(K) \subseteq O_2(K)$ for every class $K$.
A class $K$ is *closed under* $O$ when $O(K) \subseteq K$.

## Remarks

The empty product, with $I = \emptyset$, is the one-element algebra, so $P(K)$ always contains a trivial algebra.
Example: for $K = \set{\Z}$ in the signature of groups, $\Z / 2\Z \in H(K)$, by reduction modulo $2$, and $\Z^2 \in P(K)$.
Non-example: $\Z / 2\Z \notin S(K)$, because every nontrivial subgroup of $\Z$ is infinite.
Each of $H$, $S$ and $P$ is extensive, monotone and idempotent on classes closed under $I$.
The standard composition rules are $SH \le HS$, $PS \le SP$ and $PH \le HP$.