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  "body": "\n## Definition\n\nLet $\\sigma$ be an algebraic signature.\nA class $V$ of finite $\\sigma$-algebras is a *pseudovariety* if it satisfies these three conditions.\n\n1. If $A \\in V$ and $B$ is a homomorphic image of $A$, that is, isomorphic to a [[quotient algebra|N5KE7FH6CX]] of $A$, then $B \\in V$.\n2. If $A \\in V$ and $B$ is a [[subalgebra|7CHG6SD365]] of $A$, then $B \\in V$.\n3. If $A_1, \\ldots, A_n \\in V$ with $n \\ge 0$, then the [[direct product|WJ73XYBNE4]] $A_1 \\times \\cdots \\times A_n$ is in $V$; for $n = 0$ it is the one-element algebra.\n\n## Remarks\n\nExample: the class of finite abelian groups is a pseudovariety.\nQuotients and subgroups of finite abelian groups are finite and abelian, and so are finite products.\n\nNon-example: the class of finite cyclic groups fails condition 3.\nThe product $\\Z / 2\\Z \\times \\Z / 2\\Z$ is finite but not cyclic, since each of its elements has order at most $2$.\n\nFor each variety $W$, the finite members of $W$ form a pseudovariety, but not every pseudovariety arises this way.\nA standard pseudovariety of semigroups is the class of finite aperiodic semigroups, those with only trivial subgroups.\nBy 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.\n",
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