Pseudovariety

Builds on Quotient algebra Direct product of algebras

Definition

Let $\sigma$ be an algebraic signature. A class $V$ of finite $\sigma$ -algebras is a pseudovariety if it satisfies these three conditions.

If $A \in V$ and $B$ is a homomorphic image of $A$ , that is, isomorphic to a quotient algebra of $A$ , then $B \in V$ .
If $A \in V$ and $B$ is a subalgebra of $A$ , then $B \in V$ .
If $A_1, \ldots, A_n \in V$ with $n \ge 0$ , then the direct product $A_1 \times \cdots \times A_n$ is in $V$ ; for $n = 0$ it is the one-element algebra.

Remarks

Example: the class of finite abelian groups is a pseudovariety. Quotients and subgroups of finite abelian groups are finite and abelian, and so are finite products.

Non-example: the class of finite cyclic groups fails condition 3. The product $\Z / 2\Z \times \Z / 2\Z$ is finite but not cyclic, since each of its elements has order at most $2$ .

For each variety $W$ , the finite members of $W$ form a pseudovariety, but not every pseudovariety arises this way. A standard pseudovariety of semigroups is the class of finite aperiodic semigroups, those with only trivial subgroups. By Eilenberg's variety theorem, pseudovarieties of finite monoids correspond to varieties of regular languages; aperiodic monoids correspond to star-free languages by Schützenberger's theorem.

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Tag	`WV3PBEK76J`
URI	`m2://WV3PBEK76J`
Kind	definition
Writing	W0 stub
MSC	08B99, 20M07
Defines	Pseudovariety
Uses	Quotient algebra, Direct product of algebras
Changed	2 November 2026

Pseudovariety

Definition

Remarks

Reading path

Further reading

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