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  "body": "\n## Definition\n\nLet $(P, \\le_P)$ and $(Q, \\le_Q)$ be [[partially ordered sets|0QV2JS9RJ3]] with $P \\cap Q = \\varnothing$.\nThe *ordinal sum* $P \\oplus Q$ is the set $P \\cup Q$ with the relation $\\le$ given by $x \\le y$ if and only if one of the following holds:\n\n1. $x, y \\in P$ and $x \\le_P y$;\n2. $x, y \\in Q$ and $x \\le_Q y$;\n3. $x \\in P$ and $y \\in Q$.\n\n## Remarks\n\nIf $P$ and $Q$ are not disjoint, first replace them by disjoint copies.\nExample: with $P = \\set{a}$ and $Q = \\set{b, c}$ an antichain, $P \\oplus Q$ has $a \u003c b$ and $a \u003c c$ by clause 3, and $b$, $c$ stay incomparable.\nNon-example: in $Q \\oplus P$ the element $a$ is above $b$ and $c$, so $Q \\oplus P$ differs from $P \\oplus Q$; the ordinal sum is not commutative.\nThe ordinal sum is associative, and the ordinal sum of two total orders is a total order.\nFor well-orders, it gives the sum of [[ordinal numbers|0YSCSX5YZC]], so $\\omega \\oplus \\set{x}$ has order type $\\omega + 1$.\n",
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