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  "body": "\n## Summary\n\nTwo [[congruences|DCSXXT4SER]] on an [[algebra|QJGPT2W0FT]] permute when their relational products in the two orders are equal.\nThen their join in the [[congruence lattice|REE07CSXHR]] is simply their product, which makes it easy to compute.\nAn algebra whose congruences all permute is called congruence-permutable; groups, rings and modules are examples.\nThis property is the starting point of [[congruence-permutable varieties|1B3N9T68YV]] and of Mal'cev conditions.\n\n## Explanations\n\n### L1 human\n\nGiven two congruences, one can walk from $a$ to $c$ by first taking a step in one and then a step in the other.\nThe congruences permute when the order of the two steps does not matter for which pairs $(a, c)$ can be reached.\nIn a group, a step in the congruence of a normal subgroup $N$ is multiplication by an element of $N$, and two normal subgroups $N$ and $M$ satisfy $NM = MN$.\nSo all congruences of a group permute.\nFor a bare set without operations, many pairs of equivalence relations do not permute.\n",
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