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  "body": "\n## Summary\n\nA lattice congruence is an [[equivalence relation|0G8VEZ3WTK]] on a [[lattice|WH4SFAACAS]] that is compatible with [[join and meet|J2T75ATRV5]].\nIts classes can be joined and met by taking representatives, so they form a quotient lattice.\nThe congruences of a lattice are exactly the kernels of the [[lattice homomorphisms|BNVRR925X8]] defined on it.\nThis mirrors the role of normal subgroups for groups and of ideals for rings.\nThe congruences of a lattice form a distributive lattice themselves.\n\n## Explanations\n\n### L1 human\n\nA congruence groups the elements of a lattice into blocks so that the lattice operations do not see the difference inside a block.\nIf two elements lie in the same block, joining each with a third element gives two results that again lie in a common block.\nFor example, in the chain $0 \u003c 1 \u003c 2$, merge $1$ and $2$ into one block and keep $0$ alone.\nThe quotient is the two-element chain, and the map $0 \\mapsto 0$, $1 \\mapsto 1$, $2 \\mapsto 1$ collapses each block to one point.\n",
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