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  "body": "\n## Summary\n\nA modular lattice is a [[lattice|WH4SFAACAS]] that satisfies the modular law: if $a \\le b$, then $a \\vee (x \\wedge b) = (a \\vee x) \\wedge b$ for every $x$.\nThe lattices of subspaces of a vector space, of submodules of a module and of normal subgroups of a group are modular.\nEvery [[distributive lattice|0HM0HQCK6N]] is modular, but not conversely.\nModularity is what makes dimension and length behave well, as in the [[Jordan–Hölder theorem for modular lattices|BCG5KM83AP]].\n\n## Explanations\n\n### L1 human\n\nTake subspaces $A \\subseteq B$ of a vector space and any subspace $X$.\nOne can add $X$ to $A$ and then cut down to $B$, or cut $X$ down to $B$ and then add $A$; the result is the same.\nThe modular law asks for this rule in an abstract lattice.\nIt is a weak form of the distributive law, which only has to hold when two of the elements are comparable.\nThe smallest lattice that breaks it is the [[pentagon lattice N5|DHC8TZXGYS]].\n",
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