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  "body": "\n## Summary\n\nAn arithmetical variety is a [[variety|AJNZJD8442]] that is both a [[congruence-permutable variety|1B3N9T68YV]] and a [[congruence-distributive variety|9GZ7F6ETC6]].\nThe name comes from the ring of integers, whose ideals have both properties, as in the Chinese remainder theorem.\nBoolean algebras and Heyting algebras form arithmetical varieties.\nIn an arithmetical variety a form of the Chinese remainder theorem holds for congruences of every algebra.\n\n## Explanations\n\n### L1 human\n\nCongruences on the integers are given by moduli, and they combine very well: any two of them permute, and they obey the distributive law.\nAn arithmetical variety is one in which every algebra has congruences that behave this well.\nBoolean algebras are the main example.\nGroups have permuting congruences but not distributive ones, and lattices have the reverse, so neither is arithmetical.\n",
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