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  "body": "\n## Definition\n\nLet $A$ be a set.\nA partial order on $A$ is a [[preorder|0RXS8R6K37]] $\\le$ on $A$ that is [[antisymmetric|01MGYYBB9R]]. For all $a, b, c \\in A$:\n\n1. $a \\le a$;\n2. $a \\le b$ and $b \\le a$ imply $a = b$;\n3. $a \\le b$ and $b \\le c$ imply $a \\le c$.\n\nThe pair $(A, \\le)$ is a partially ordered set.\n\n## Remarks\n\nExamples are $\\le$ on $\\mathbb{R}$, divisibility on $\\mathbb{N}$, and $\\subseteq$ on the [[power set|0BC7KX51Z8]] of a set.\n\"Partial\" means that two elements need not be comparable: neither $\\{1\\} \\subseteq \\{2\\}$ nor $\\{2\\} \\subseteq \\{1\\}$.\nThe strict order $a \u003c b$ means $a \\le b$ and $a \\neq b$.\nA finite poset can be drawn as a [[Hasse diagram|00EYM7MDKS]].\n\n## Explanations\n\n### L1 human\n\nA partial order ranks some pairs of elements, but not necessarily all pairs.\nIn the family tree, \"is an ancestor of or equal to\" is a partial order: two cousins are not comparable.\n",
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