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  "body": "\n## Statement\n\nLet $A$ be a finite algebra of a finite signature with only function symbols.\nAssume that the [[variety|AJNZJD8442]] generated by $A$ is [[congruence-distributive|9GZ7F6ETC6]].\nThen $A$ is [[finitely based|YGYZAQ12MZ]].\n\n## Remarks\n\nKirby Baker proved the theorem in 1977.\nOne ingredient is Jónsson's lemma, which shows that the subdirectly irreducible algebras in the variety generated by $A$ are quotients of subalgebras of $A$, so they are finite and few.\nThe other ingredient is the Jónsson terms of a congruence-distributive variety, which Baker used to find finitely many identities that axiomatise the variety.\n\nExample: every finite lattice is finitely based, since lattices form a congruence-distributive variety.\nThe same holds for every finite Boolean algebra and every finite Heyting algebra, since these have lattice reducts and so generate congruence-distributive varieties.\n\nThe finite signature is needed.\nLet $A$ be the two-element lattice $\\set{0, 1}$ with infinitely many extra unary operations $f_1, f_2, \\ldots$, all equal to the identity map.\nIt has a lattice reduct, so it generates a congruence-distributive variety, and it satisfies $f_i(x) \\approx x$ for every $i$.\nA finite set $\\Sigma$ of identities of $A$ mentions only the $f_i$ with $i$ in some finite set $S$.\nChange $A$ by letting $f_j$ be the constant map with value $0$ for every $j \\notin S$.\nThe new algebra still satisfies $\\Sigma$, but it fails $f_j(x) \\approx x$ at $x = 1$, so $\\Sigma$ is not an equational basis of $A$.\n\n## Explanations\n\n### L1 human\n\nThe theorem gives a large family of finite algebras whose laws follow from finitely many of them.\nIt applies whenever the congruences of the algebras in the variety behave like a distributive lattice, as they do for lattices and Boolean algebras.\nSo every finite lattice can be described by finitely many laws.\n\n### L2 human\n\nBaker's theorem is one of the main positive answers to Tarski's finite basis problem.\nIt extends McKenzie's earlier result that finite lattices are finitely based.\nIt was later generalised by McKenzie to congruence-modular varieties with a finite residual bound, and by Willard to congruence meet-semidistributive varieties with a finite residual bound.\nThese results are steps towards Park's conjecture, which predicts that a finite algebra of finite signature that generates a variety with a finite residual bound is finitely based.\n",
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