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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]], with strict order $\u003c$.\nThe poset $P$ is an *interval order* if there are real numbers $l_x \\le r_x$ for each $x \\in P$ such that, for all $x, y \\in P$,\n$$x \u003c y \\iff r_x \u003c l_y.$$\nThe closed [[interval|0Q1D85B8G6]] $[l_x, r_x]$ is said to represent $x$.\n\n## Remarks\n\nExample: the poset on $\\set{a, b, c}$ with $a \u003c b$ as the only strict relation is an interval order.\nThe intervals $[0, 1]$, $[2, 3]$ and $[0.5, 2.5]$ for $a$, $b$, $c$ work: $1 \u003c 2$ gives $a \u003c b$, and the interval of $c$ meets both others.\nNon-example: the poset $\\mathbf{2} + \\mathbf{2}$ with $a \u003c b$, $c \u003c d$ and no other strict relation is not an interval order.\nIntervals would give $r_a \u003c l_b$ and $r_c \u003c l_d$, and also $l_d \\le r_a$ and $l_b \\le r_c$, so $r_a \u003c l_b \\le r_c \u003c l_d \\le r_a$, which is impossible.\nFishburn's theorem says that a finite poset is an interval order if and only if it has no subposet isomorphic to $\\mathbf{2} + \\mathbf{2}$.\nEvery total order is an interval order: on a finite total order, use the points $[i, i]$.\n",
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