Class operators H S P

Builds on Homomorphism Isomorphism Direct product of algebras

Definition

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

$I(K)$ is the class of algebras isomorphic to a member of $K$ .
$H(K)$ is the class of algebras $B$ such that there is a surjective homomorphism $A \to B$ for some $A \in K$ .
$S(K)$ is the class of algebras isomorphic to a subalgebra of a member of $K$ .
$P(K)$ is the class of algebras isomorphic to a direct product $\prod_{i \in I} A_i$ 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 $PH \le HP$ .

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Tag	`J037NW1Q38`
URI	`m2://J037NW1Q38`
Kind	definition
Writing	W0 stub
MSC	08B99, 03C05
Defines	Class operators H S P
Uses	Homomorphism, Isomorphism, Direct product of algebras
Changed	2 November 2026

Class operators H S P

Definition

Remarks

Reading path

Further reading

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