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  "body": "\n## Definition\n\nA lattice $(L, \\vee, \\wedge)$ is distributive if $x \\wedge (y \\vee z) = (x \\wedge y) \\vee (x \\wedge z)$ for all $x, y, z \\in L$.\n\n## Remarks\n\nThe condition is equivalent to its dual: $x \\vee (y \\wedge z) = (x \\vee y) \\wedge (x \\vee z)$ for all $x, y, z$.\nEvery totally ordered set is a distributive lattice, with $\\max$ as join and $\\min$ as meet. Every [[Boolean algebra|06GKBTAA5E]] is a distributive lattice.\nThe diamond lattice $M_3$ and the pentagon lattice $N_5$ are the basic lattices that are not distributive.\n\n## Explanations\n\n### L1 human\n\nA lattice is distributive when meet distributes over join, like intersection over union for sets.\nThen join also distributes over meet.\n",
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