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  "body": "\n## Example\n\nLet $n \\ge 1$ and let $\\Pi_n$ be the set of all [[partitions|08QMKTWF9T]] of $\\set{1, \\ldots, n}$.\nOrder $\\Pi_n$ by [[refinement|0SPZ1H3N9V]]: $\\sigma \\le \\tau$ when each block of $\\sigma$ lies inside some block of $\\tau$.\nThis is a partial order, and $\\Pi_n$ is a [[lattice|WH4SFAACAS]].\nThe meet $\\sigma \\wedge \\tau$ has as blocks the nonempty intersections $B \\cap C$ of a block $B$ of $\\sigma$ and a block $C$ of $\\tau$.\nThe join $\\sigma \\vee \\tau$ is the finest partition coarser than both.\nIn it, two points share a block when a chain of blocks of $\\sigma$ and $\\tau$, each meeting the next, connects them.\nThe least element is the partition into singletons, and the greatest element is the partition with one block.\nThe number of elements of $\\Pi_n$ is the [[Bell number|0TKP3TDD02]] $B_n$.\n\nCheck for $n = 3$.\nThe five partitions are $1|2|3$, $12|3$, $13|2$, $23|1$ and $123$.\nThe three partitions with one pair lie strictly between the bottom and the top, and no two of them are comparable.\nThe meet of any two of them is $1|2|3$ and their join is $123$, so $\\Pi_3$ is the [[diamond lattice|JQ6FJH2TP8]] $M_3$.\n\n## Remarks\n\nThe lattice $\\Pi_n$ is graded: the rank of a partition with $k$ blocks is $n - k$.\nIt is a [[geometric lattice|VDF7CGPC0P]], but for $n \\ge 3$ it is not distributive, since it contains $M_3$.\nThe Möbius function of $\\Pi_n$ from the bottom to the top equals $(-1)^{n-1} (n-1)!$.\n\n## Explanations\n\n### L1 human\n\nThink of ways to split a group of $n$ people into teams.\nOne splitting is below another when you get the second by merging some teams of the first.\nAny two splittings have a best common refinement and a coarsest common merge, so the splittings form a lattice.\nFor three people there are five splittings, and the three ways to form one pair sit side by side in the middle.\n",
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