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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] with identity $e$, and let $\\le$ be a [[partial order|011C29Q1S1]] on $G$.\nThe pair $(G, \\le)$ is an *ordered group*, or *partially ordered group*, if for all $a, b, c \\in G$,\n$$a \\le b \\implies ca \\le cb \\text{ and } ac \\le bc.$$\n\n## Remarks\n\nExample: $\\Z^2$ under addition, with $(a, b) \\le (c, d)$ when $a \\le c$ and $b \\le d$, is an ordered group.\nIf $a \\le c$ and $b \\le d$, then $a + x \\le c + x$ and $b + y \\le d + y$ for each $(x, y) \\in \\Z^2$.\nNon-example: on $\\Z$, let $a \\preceq b$ when $a = b$ or $(a, b) = (0, 1)$.\nThis is a partial order, but $0 \\preceq 1$ while $1 \\not\\preceq 2$, so adding $1$ breaks the condition.\nThe *positive cone* of $G$ is $P = \\set{g \\in G : e \\le g}$.\nIt satisfies $PP \\subseteq P$, $P \\cap P^{-1} = \\set{e}$ and $gPg^{-1} = P$ for each $g \\in G$.\nConversely, each subset $P$ with these three properties defines an ordered group by $a \\le b \\iff a^{-1} b \\in P$.\nFor an abelian group the condition is written additively: $a \\le b$ implies $a + c \\le b + c$.\n",
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    "name": "Ordered group",
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