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  "body": "\n## Summary\n\nThe Krull dimension of a [[partially ordered set|0QV2JS9RJ3]] is the supremum of the lengths of its finite [[chains|0S8FE7ZMFM]].\nHere a chain with $n + 1$ elements has length $n$.\nIt generalizes the [[Krull dimension of a commutative ring|MVGBS3J1P7]], which is the Krull dimension of the set of prime ideals ordered by inclusion.\nFor a nonempty finite poset it is one less than the [[height|MPR6MH4YSQ]] counted in elements.\n\n## Explanations\n\n### L1 human\n\nCount the number of upward steps in the longest climb through the poset.\nA single point has dimension $0$, a two-element chain has dimension $1$, and the subsets of a set with $n$ elements have dimension $n$.\nThe name comes from algebra, where a point, a curve and a surface give chains of prime ideals with $0$, $1$ and $2$ steps.\n",
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