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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|011C29Q1S1]].\nThe poset $P$ is *bounded* when it has a [[greatest element|0Z7HDDXX3X]] $\\top$ and a [[least element|0T2NTJCV83]] $\\bot$, that is, $\\bot \\le x \\le \\top$ for all $x \\in P$.\n\n## Remarks\n\nThe positive divisors of $12$, ordered by divisibility, form a bounded poset with $\\bot = 1$ and $\\top = 12$, since $1 \\mid d$ and $d \\mid 12$ for each divisor $d$.\nThe poset $(\\Z, \\le)$ is not bounded: it has neither a greatest nor a least element.\nThe interval $(0, 1]$ in $\\R$ has the top $1$ but no bottom, so it is not bounded.\nFor a subset of $P$, being bounded usually means something weaker: having an [[upper bound|0H5J02P8S9]] and a lower bound in $P$.\nThe elements $\\top$ and $\\bot$ are unique when they exist, by antisymmetry.\nA bounded poset with one element has $\\top = \\bot$.\n\n## Explanations\n\n### L1 human\n\nA bounded poset has a top and a bottom.\nEvery element lies between the two.\n",
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