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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|011C29Q1S1]] and let $S$ be a [[subset|0FE8Q5S42Y]] of $P$.\nAn element $m \\in S$ is a maximal element of $S$ when no $s \\in S$ satisfies $m \u003c s$.\nEquivalently, for each $s \\in S$, $m \\le s$ implies $s = m$.\n\n## Remarks\n\nA maximal element need not be the [[greatest element|0Z7HDDXX3X]], and a set can have several maximal elements.\nIn $\\{\\{1\\}, \\{2\\}, \\{1, 2, 3\\}, \\{4\\}\\}$ under $\\subseteq$, the maximal elements are $\\{1, 2, 3\\}$ and $\\{4\\}$.\n[[Zorn's lemma|04RXTQA5ZA]] gives maximal elements in many infinite posets.\n\n## Explanations\n\n### L1 human\n\nA maximal element has nothing above it.\nIt may still be incomparable with other elements, so it is not necessarily above everything.\n",
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