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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]].\nA *left order* on $G$ is a [[total order|0316AYH1SC]] $\\le$ on $G$ such that for all $a, b, c \\in G$,\n$$a \\le b \\implies ca \\le cb.$$\nThe group $G$ is *left-orderable* if it has a left order.\n\n## Remarks\n\nEvery [[orderable group|ZZFFBXDTWN]] is left-orderable.\nExample: the Klein bottle group $K = \\langle a, b \\mid b a b^{-1} = a^{-1} \\rangle$ is left-orderable but not orderable.\nEvery element of $K$ is $b^n a^m$ for unique integers $n, m$.\nSet $b^n a^m \\le b^{n'} a^{m'}$ when $n \u003c n'$, or $n = n'$ and $m \\le m'$.\nLeft multiplication by $b$ adds $1$ to $n$, and left multiplication by $a$ sends $b^n a^m$ to $b^n a^{m + (-1)^n}$.\nBoth keep this order, so it is a left order.\nIn a two-sided order, $a \u003e e$ would give $a^{-1} = b a b^{-1} \u003e e$, which contradicts $a \u003e e$.\nThe case $a \u003c e$ is the same, so $K$ is not orderable.\nNon-example: a group with an element $g \\ne e$ of finite order $n$ is not left-orderable, since $e \u003c g$ gives $e \u003c g \u003c g^2 \u003c \\cdots \u003c g^n = e$.\nIf $g \u003c e$, the same argument applies to $g^{-1}$.\nSo a left-orderable group is a [[torsion-free group|VQ4SSKS53D]].\nEquivalently, $G$ is left-orderable if and only if it has a subsemigroup $P$ with $G = P \\sqcup \\set{e} \\sqcup P^{-1}$.\n",
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    "name": "Left-orderable group",
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