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  "body": "\n## Statement\n\nLet $\\sigma$ be a finite algebraic signature and let $\\mathcal{F}$ be the class of all finite $\\sigma$-algebras.\nFor $n \\ge 1$, an *$n$-ary implicit operation* $\\pi$ assigns to each $A \\in \\mathcal{F}$ a map $\\pi_A \\colon A^n \\to A$, subject to one condition.\nFor every [[homomorphism|KBXSZC6BV1]] $f \\colon A \\to B$ in $\\mathcal{F}$ and all $a_1, \\ldots, a_n \\in A$, it requires $f(\\pi_A(a_1, \\ldots, a_n)) = \\pi_B(f(a_1), \\ldots, f(a_n))$.\nA *pseudoidentity* is a pair $\\pi \\approx \\rho$ of $n$-ary implicit operations, and $A \\in \\mathcal{F}$ satisfies it if $\\pi_A = \\rho_A$.\nLet $V \\subseteq \\mathcal{F}$.\nThen $V$ is a [[pseudovariety|XR4DPKX7T1]] if and only if there is a set $\\Sigma$ of pseudoidentities with the following property.\nAn algebra $A \\in \\mathcal{F}$ is in $V$ exactly when it satisfies every member of $\\Sigma$.\n\n## Remarks\n\nEach term gives an implicit operation, so every [[identity|PKD440G2XW]] is a pseudoidentity, but there are many more.\nThe $n$-ary implicit operations are the elements of the free profinite $\\sigma$-algebra on $n$ generators, the inverse limit of all $n$-generated finite $\\sigma$-algebras.\nThe easy direction holds because satisfaction of a pseudoidentity passes to quotients, subalgebras and finite products.\nFor the converse, one takes as $\\Sigma$ all pseudoidentities true in $V$.\nA finite algebra $A$ satisfying them is a continuous image of a free pro-$V$ algebra.\nBy compactness this map factors through a finite quotient of it that lies in $V$, so $A \\in V$.\n\nExample: in finite semigroups, each element $x$ has a unique idempotent power, written $x^\\omega$; it is the limit of $x^{n!}$, and $x \\mapsto x^\\omega$ is a unary implicit operation.\nThe pseudovariety of finite groups is defined by $x^\\omega y \\approx y$ and $y x^\\omega \\approx y$.\nThe pseudovariety of finite aperiodic semigroups is defined by $x^\\omega \\approx x^{\\omega + 1}$.\nNeither class is the class of finite members of a variety of semigroups, so ordinary identities do not suffice.\n\n## Explanations\n\n### L1 human\n\nBirkhoff's theorem says that classes of algebras closed under the usual constructions are exactly those defined by equations.\nFor finite algebras the equations are not enough.\nReiterman showed that they become enough once one allows \"limit\" operations, such as taking the idempotent power of an element, which make sense in every finite algebra.\n\n### L2 human\n\nThe theorem is the finite counterpart of [[Birkhoff's HSP theorem|5083DHSK82]].\nIt is a basic tool in the algebraic theory of regular languages: through Eilenberg's correspondence, a class of regular languages is described by pseudoidentities of its syntactic monoids.\nDecidability questions for pseudovarieties then become questions about the free profinite semigroup or monoid.\nAn earlier description, due to Eilenberg and Schützenberger, defines each pseudovariety by a sequence of identities that is eventually satisfied.\n",
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