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  "body": "\n## Definition\n\nLet $(L, \\vee, \\wedge)$ be a [[lattice|WH4SFAACAS]].\nA *congruence* on $L$ is an [[equivalence relation|0G8VEZ3WTK]] $\\theta$ on $L$ such that, for all $x, y, x', y' \\in L$ with $x \\mathrel{\\theta} y$ and $x' \\mathrel{\\theta} y'$:\n\n1. $(x \\vee x') \\mathrel{\\theta} (y \\vee y')$;\n2. $(x \\wedge x') \\mathrel{\\theta} (y \\wedge y')$.\n\n## Remarks\n\nThe quotient set $L / \\theta$ is a lattice with $[x] \\vee [y] = [x \\vee y]$ and $[x] \\wedge [y] = [x \\wedge y]$; clauses 1 and 2 say that these operations are well defined.\nThe kernel $\\set{(x, y) : f(x) = f(y)}$ of a [[lattice homomorphism|BNVRR925X8]] $f$ is a congruence, and every congruence $\\theta$ is the kernel of the quotient map $L \\to L / \\theta$.\nEach class of a congruence is a convex sublattice of $L$.\nExample: on the chain $0 \u003c 1 \u003c 2$, the partition $\\set{0}, \\set{1, 2}$ is a congruence.\nIt is the kernel of the monotone map $0 \\mapsto 0$, $1 \\mapsto 1$, $2 \\mapsto 1$ onto the chain $0 \u003c 1$.\nThis map is a lattice homomorphism because join and meet in a chain are $\\max$ and $\\min$.\nNon-example: on the lattice $\\set{0, a, b, 1}$ with $0 \u003c a, b \u003c 1$ and $a$, $b$ incomparable, the partition $\\set{0, 1}, \\set{a}, \\set{b}$ is not a congruence.\nFrom $0 \\mathrel{\\theta} 1$ and $a \\mathrel{\\theta} a$, clause 1 would give $(0 \\vee a) \\mathrel{\\theta} (1 \\vee a)$, that is $a \\mathrel{\\theta} 1$, which is false.\n",
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