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  "body": "\n## Summary\n\nA maximal chain in a [[partially ordered set|011C29Q1S1]] is a [[chain|0GP9ASVQ1X]] that is not contained in any strictly larger chain.\nThe [[Hausdorff maximal principle|0KM4KVR9MV]] says that every chain lies in a maximal chain, and this statement is equivalent to the axiom of choice.\nIn finite posets, maximal chains are used to define the height and the grading of a poset.\n\n## Explanations\n\n### L1 human\n\nA chain is a set of elements that stand on one line in the order.\nA maximal chain is a chain that cannot be extended: no other element can be added while keeping all elements comparable.\nIn the subsets of $\\set{1, 2}$, the chain $\\emptyset \\subset \\set{1} \\subset \\set{1, 2}$ is maximal.\nThe chain $\\emptyset \\subset \\set{1, 2}$ is not maximal, because $\\set{1}$ can be inserted.\n",
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