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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]].\nThe *height* of $P$ is the supremum of the cardinalities $\\abs{C}$ of the [[chains|0S8FE7ZMFM]] $C \\subseteq P$.\n\n## Remarks\n\nExample: the subsets of $\\set{1, 2, 3}$ ordered by inclusion have height $4$, as shown by the chain $\\varnothing \\subset \\set{1} \\subset \\set{1, 2} \\subset \\set{1, 2, 3}$.\nA chain of subsets has at most one subset of each size, so no chain is longer.\nNon-example: the set $\\set{\\varnothing, \\set{1}, \\set{2}, \\set{1, 2}}$ has $4$ elements but is not a chain, since $\\set{1}$ and $\\set{2}$ are incomparable.\nConventions differ: some authors call the height the largest *length* of a chain $x_0 \u003c x_1 \u003c \\cdots \u003c x_n$, which is $n$, one less than the number of elements.\nIn that second convention the height is the Krull dimension of the poset.\nThe height of an element $x$ is often defined as the height of the down-set $\\set{y \\in P : y \\le x}$, with the same choice of convention.\n",
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