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  "body": "\n## Definition\n\nLet $L$ be a [[lattice|WH4SFAACAS]] with [[join and meet|J2T75ATRV5]] $\\vee$ and $\\wedge$.\nWrite $x \\lessdot y$ when $y$ [[covers|4HHDPZ4BDY]] $x$, that is, $x \u003c y$ and no $z \\in L$ satisfies $x \u003c z \u003c y$.\nThe lattice $L$ is *semimodular*, or *upper semimodular*, if for all $a, b \\in L$,\n$$a \\wedge b \\lessdot a \\implies b \\lessdot a \\vee b.$$\nDually, $L$ is *lower semimodular* if $b \\lessdot a \\vee b$ implies $a \\wedge b \\lessdot a$ for all $a, b \\in L$.\n\n## Remarks\n\nEvery [[modular lattice|QX7WZ92QAK]] is both upper and lower semimodular, because in a modular lattice the intervals $[a \\wedge b, a]$ and $[b, a \\vee b]$ are isomorphic.\n\nExample: the [[lattice of partitions|9RP82GD0Q4]] of a finite set, ordered by refinement with the coarser partition above, is upper semimodular.\nIf a partition $a$ covers $a \\wedge b$, then $a$ arises from $a \\wedge b$ by merging two blocks.\nThen $a \\vee b$ arises from $b$ by merging the two blocks of $b$ that contain these two blocks.\nThese blocks of $b$ are different, since otherwise $a \\le b$ and $a = a \\wedge b$; so $b \\lessdot a \\vee b$.\nFor a set of $4$ or more elements, this lattice is not modular.\n\nNon-example: the [[pentagon lattice N5|DHC8TZXGYS]] with $0 \u003c a \u003c c \u003c 1$ and $b$ incomparable to $a$ and $c$.\nApply the condition to the pair $(b, a)$: $b \\wedge a = 0$ and $0 \\lessdot b$, but $a \u003c c \u003c 1 = a \\vee b$, so $a \\lessdot a \\vee b$ fails.\n\nFor a finite lattice, upper semimodularity holds if and only if the lattice is graded with a rank function $r$ satisfying $r(a) + r(b) \\ge r(a \\wedge b) + r(a \\vee b)$ for all $a, b$.\n",
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    "name": "Semimodular lattice",
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        "url": "https://en.wikipedia.org/wiki/Semimodular_lattice"
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