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  "body": "\n## Statement\n\nLet $P$ be a finite poset on a set $X$ that contains no $\\mathbf{2}+\\mathbf{2}$. Let $\\mathcal{D}$ be the family of distinct strict down sets $\\{y : y \u003c x\\}$ of elements $x \\in X$. Let $\\mathcal{U}$ be the family of distinct strict up sets $\\{y : y \u003e x\\}$ (see [[upper set|ETCE287E2R]]). Both families are chains under inclusion. Order the sets of each family and use their positions to give every $x \\in X$ an interval $I(x)$ with integer end points. This is the standard [[interval order|VSNQV29KFJ]] representation of $P$. Then all of these hold.\n\n1. The two families have the same size: $|\\mathcal{D}| = |\\mathcal{U}|$. Call this number $d$.\n2. For each $x \\in X$ with $I(x) = [i, j]$, we have $i \\le j$.\n3. For $x, y \\in X$ with $I(x) = [i, j]$ and $I(y) = [k, l]$, we have $x \u003c y$ in $P$ if and only if $j \u003c k$.\n4. The number $d$ is the least positive integer such that $P$ has an interval representation with all end points in $\\{1, \\dots, d\\}$. This representation is unique.\n\n## Remarks\n\nParts 2 and 3 say that the intervals $I(x)$ are genuine intervals and that they encode the order of $P$ exactly. Part 4 says that this representation uses the fewest possible integer values and that nothing else does as well.\n",
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        "url": "https://github.com/mitchkeller/applied-combinatorics/blob/ebd95d6bd59e23c633c65cf560bcb93efa912aae/source/s_posets_intervalorder_findrep.ptx#L53"
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  "msc": [
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  "name": "Minimal integer representation of an interval order",
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