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  "body": "\n## Statement\n\nLet $L$ be a [[lattice|WH4SFAACAS]].\nLet $M_3$ be the [[diamond lattice|JQ6FJH2TP8]] and $N_5$ the [[pentagon lattice|DHC8TZXGYS]].\nThen $L$ is a [[distributive lattice|0HM0HQCK6N]] if and only if $L$ has no [[sublattice|PV90EW00XK]] isomorphic to $M_3$ and no sublattice isomorphic to $N_5$.\n\n## Remarks\n\nIf $L$ is distributive, so is every sublattice, and $M_3$ and $N_5$ are not distributive.\nConversely, if $L$ has no copy of $N_5$, then $L$ is [[modular|QX7WZ92QAK]] by [[Dedekind's characterization|ANRXAG227H]].\nIn a modular lattice in which $a \\wedge (b \\vee c) \\ne (a \\wedge b) \\vee (a \\wedge c)$, put $d = (a \\wedge b) \\vee (b \\wedge c) \\vee (c \\wedge a)$ and $e = (a \\vee b) \\wedge (b \\vee c) \\wedge (c \\vee a)$.\nThen $d \u003c e$, and the elements $d$, $e$, $(a \\wedge e) \\vee d$, $(b \\wedge e) \\vee d$, $(c \\wedge e) \\vee d$ form a sublattice isomorphic to $M_3$.\n\nExample: the lattice of subspaces of $\\R^2$ is not distributive, because the zero subspace, three distinct lines and $\\R^2$ form a copy of $M_3$.\nThe lattice of divisors of $12$ under divisibility, with $\\gcd$ as meet and $\\lcm$ as join, is distributive, so it contains neither shape.\n\n## Explanations\n\n### L1 human\n\nThe distributive law is the familiar rule that \"and\" distributes over \"or\".\nThe theorem says that a lattice breaks this rule only if one of two small five-element lattices, the diamond or the pentagon, sits inside it.\nSo one can test distributivity by looking for these two shapes.\n\n### L2 human\n\nThis theorem is the standard way to show that a lattice is not distributive: one exhibits a diamond or a pentagon.\nIt refines [[Dedekind's characterization|ANRXAG227H]] of modular lattices, which forbids only the pentagon.\nFor a finite distributive lattice, the structure is then described completely by [[Birkhoff's representation theorem|6RDM3069KC]].\nThe proof constructs the forbidden sublattice explicitly from any triple that violates distributivity.\n",
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    "name": "M3–N5 characterization of distributivity",
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        "url": "https://en.wikipedia.org/wiki/Distributive_lattice"
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  "markdown": "https://tamnd.github.io/math2/node/C6GH4VHAMQ.md",
  "msc": [
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    "06C05"
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  "name": "M3–N5 characterization of distributivity",
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  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/C6G/C6GH4VHAMQ.md",
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