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  "body": "\n## Definition\n\nLet $X$ be a set and let $\\prec$ be a [[strict partial order|60FNZ35H6R]] on $X$.\nSay that $x$ and $y$ are incomparable when neither $x \\prec y$ nor $y \\prec x$.\nThe order $\\prec$ is a *semiorder* if, for all $a, b, c, d \\in X$:\n\n1. if $a \\prec b$ and $c \\prec d$, then $a \\prec d$ or $c \\prec b$;\n2. if $a \\prec b$ and $b \\prec c$, then $a \\prec d$ or $d \\prec c$.\n\n## Remarks\n\nClause 1 forbids two incomparable two-element chains, and clause 2 forbids a three-element chain with an element incomparable to all three of its elements.\nExample: on $\\R$, let $x \\prec y$ when $x + 1 \u003c y$.\nFor clause 2, if $a + 1 \u003c b$ and $b + 1 \u003c c$ but $d \\le a + 1$, then $d + 1 \\le a + 2 \u003c c$, so $d \\prec c$; clause 1 has a similar check.\nNon-example: the poset with $a \\prec b \\prec c$ and an element $d$ incomparable to $a$, $b$ and $c$ satisfies clause 1 but fails clause 2.\nEvery semiorder is an [[interval order|VSNQV29KFJ]].\nBy a theorem of Scott and Suppes, a finite strict partial order is a semiorder if and only if there is a map $u \\colon X \\to \\R$ with $x \\prec y \\iff u(x) + 1 \u003c u(y)$.\n",
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