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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]].\nThe *width* of $P$ is the supremum of the cardinalities $\\abs{A}$ of the [[antichains|03HF498P1J]] $A \\subseteq P$.\n\n## Remarks\n\nExample: the set $\\set{1, \\ldots, 10}$ ordered by divisibility has width $5$.\nThe set $\\set{6, 7, 8, 9, 10}$ is an antichain, and the five chains $\\set{1, 2, 4, 8}$, $\\set{3, 6}$, $\\set{5, 10}$, $\\set{7}$, $\\set{9}$ cover the set, so no antichain has more than $5$ elements.\nNon-example: the set $\\set{2, 3, 4}$ has $3$ elements but is not an antichain there, since $2 \\mid 4$, so it does not show that the width is at least $3$.\nA nonempty poset has width $1$ exactly when it is a total order.\nThe width of an infinite poset can be infinite even when each antichain is finite.\n[[Dilworth's theorem|GT224XDA4E]] says that a finite poset of width $w$ is a union of $w$ chains and of no fewer.\n",
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