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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 *disjoint union* $P + Q$ is the set $P \\cup Q$ with the following relation $\\le$.\nFor $x, y \\in P \\cup Q$, $x \\le y$ holds if and only if either $x, y \\in P$ and $x \\le_P y$, or $x, y \\in Q$ and $x \\le_Q y$.\n\n## Remarks\n\nIf $P$ and $Q$ are not disjoint, first replace them by disjoint copies, such as $P \\times \\set{0}$ and $Q \\times \\set{1}$.\nThe same construction works for any family of pairwise disjoint posets.\nExample: if $P$ is the chain $a \u003c b$ and $Q$ is the chain $c \u003c d$, then $P + Q$ is the poset $\\mathbf{2} + \\mathbf{2}$ with exactly the strict relations $a \u003c b$ and $c \u003c d$.\nNon-example: the poset with $a \u003c b$, $c \u003c d$ and also $a \u003c d$ is not $P + Q$, since it relates an element of $P$ to an element of $Q$.\nAn antichain with $n$ elements is the disjoint union of $n$ one-element posets.\nThe width of $P + Q$ is the sum of the widths, and its height is the larger of the two heights.\n",
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