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  "body": "\n## Definition\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras.\nThe variety $\\mathcal{V}$ is *arithmetical* if it is both a [[congruence-permutable variety|1B3N9T68YV]] and a [[congruence-distributive variety|9GZ7F6ETC6]].\n\n## Remarks\n\nEquivalently, $\\mathcal{V}$ is arithmetical if and only if it has a Pixley term $p$, with $p(x, y, y) = x$, $p(x, x, y) = y$ and $p(x, y, x) = x$.\nExample: the variety of Boolean algebras is arithmetical.\nIt is congruence distributive because Boolean algebras are lattices, and it is congruence permutable because the term $p(x, y, z) = (x \\wedge \\neg y) \\vee (x \\wedge z) \\vee (\\neg y \\wedge z)$ is a Mal'cev term.\nNon-example: the variety of groups is congruence permutable but not congruence distributive, so it is not arithmetical.\nNon-example: the variety of lattices is congruence distributive but not congruence permutable.\n\n## Explanations\n\n### L1 human\n\nA variety is arithmetical when the congruences of each of its algebras both permute and satisfy the distributive law, as the congruences of the integers do.\n",
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