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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|011C29Q1S1]].\nThe *dual* of $P$ is the poset $P^{\\mathrm{op}} = (P, \\ge)$ on the same set, where $x \\ge y$ means $y \\le x$.\nThe *dual* of a statement about partial orders is the statement obtained by replacing $\\le$ with $\\ge$ throughout.\nThen each defined notion is replaced by its dual notion, such as [[upper bound|0H5J02P8S9]] by lower bound and [[greatest element|0Z7HDDXX3X]] by [[least element|0T2NTJCV83]].\n\n## Remarks\n\nThe relation $\\ge$ is again a partial order, since reflexivity, antisymmetry and transitivity are symmetric in the two sides.\nFor example, the dual of $(\\N, \\le)$ is $\\N$ with the reverse order, in which $0$ is the greatest element and there is no least element.\nThe dual of the dual of $P$ is $P$ itself.\nDuality principle: if a statement holds in every poset, then its dual statement holds in every poset.\nThe reason is that the dual statement for $P$ is the original statement for $P^{\\mathrm{op}}$.\nThe dual of $P$ is also written $P^{\\partial}$ or $P^{*}$.\n\n## Explanations\n\n### L1 human\n\nThe dual order reads the same order backwards.\nA theorem about orders, read backwards, is again a theorem.\n",
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