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  "body": "\n## Definition\n\nLet $(L, \\vee, \\wedge)$ and $(M, \\vee, \\wedge)$ be [[lattices|WH4SFAACAS]].\nA *lattice homomorphism* is a map $f \\colon L \\to M$ such that, for all $x, y \\in L$:\n\n1. $f(x \\vee y) = f(x) \\vee f(y)$;\n2. $f(x \\wedge y) = f(x) \\wedge f(y)$.\n\nIf $L$ and $M$ are bounded lattices, a *bounded lattice homomorphism* also satisfies $f(0) = 0$ and $f(1) = 1$.\n\n## Remarks\n\nA lattice homomorphism is [[monotone|0A7KV2WB7A]]: if $x \\le y$ then $x \\vee y = y$, so $f(x) \\vee f(y) = f(y)$, that is, $f(x) \\le f(y)$.\nA bijective lattice homomorphism is a lattice isomorphism, and its inverse is again a lattice homomorphism.\nA bijection between lattices is a lattice isomorphism exactly when it and its inverse are monotone.\nExample: every monotone map between two chains is a lattice homomorphism, since join and meet in a chain are $\\max$ and $\\min$, and monotone maps preserve both.\nNon-example: let $N = \\set{\\emptyset, \\set{1}, \\set{2}, \\set{1, 2, 3}}$, a lattice under $\\subseteq$ in which $\\set{1} \\vee \\set{2} = \\set{1, 2, 3}$.\nThe inclusion map of $N$ into the power set of $\\set{1, 2, 3}$ is monotone and preserves meets.\nIt fails clause 1, because the join of $\\set{1}$ and $\\set{2}$ in the power set is $\\set{1, 2} \\ne \\set{1, 2, 3}$.\n",
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