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  "body": "\n## Definition\n\nLet $(L, \\vee, \\wedge)$ be a [[lattice|WH4SFAACAS]] with [[join and meet|J2T75ATRV5]] $\\vee$ and $\\wedge$ and order $\\le$.\nThe lattice $L$ is *modular* if for all $a, b, x \\in L$ with $a \\le b$,\n$$a \\vee (x \\wedge b) = (a \\vee x) \\wedge b.$$\n\n## Remarks\n\nThe inequality $a \\vee (x \\wedge b) \\le (a \\vee x) \\wedge b$ holds in every lattice when $a \\le b$, so only the reverse inequality is a condition.\nEquivalently, $L$ satisfies the identity $((a \\wedge b) \\vee x) \\wedge b = (a \\wedge b) \\vee (x \\wedge b)$ for all $a, b, x \\in L$, so the modular lattices form a variety.\n\nExample: the lattice of subspaces of a vector space $V$ is modular.\nIf $A \\subseteq B$ and $X$ are subspaces, and $b = a + x \\in (A + X) \\cap B$ with $a \\in A$, $x \\in X$, then $x = b - a \\in B$, so $b \\in A + (X \\cap B)$.\n\nNon-example: the [[pentagon lattice N5|DHC8TZXGYS]] with $0 \u003c a \u003c c \u003c 1$ and $b$ incomparable to $a$ and $c$.\nHere $a \\le c$, but $a \\vee (b \\wedge c) = a \\vee 0 = a$ while $(a \\vee b) \\wedge c = 1 \\wedge c = c$.\n\nEvery [[distributive lattice|0HM0HQCK6N]] is modular.\nThe [[diamond lattice M3|JQ6FJH2TP8]] is modular but not distributive.\n",
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