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  "body": "\n## Definition\n\nLet $(G, +, \\le)$ be a [[linearly ordered group|1P8087YGYQ]] that is an [[abelian group|071DDFVG4J]].\nFor $a \\in G$, let $\\abs{a} = \\max(a, -a)$.\nTwo elements $a, b \\in G$ are *Archimedean equivalent*, written $a \\sim b$, if there are integers $m, n \\ge 1$ with\n$$\\abs{a} \\le n \\abs{b} \\quad \\text{and} \\quad \\abs{b} \\le m \\abs{a}.$$\nThis is an equivalence relation on $G$.\nIts classes are the *Archimedean classes* of $G$, and the class of $0$ is $\\set{0}$.\nFor nonzero $a, b$ with $a \\not\\sim b$, one says that $a$ is *infinitely larger* than $b$ if $n \\abs{b} \u003c \\abs{a}$ for all $n \\ge 1$.\n\n## Remarks\n\nExactly one of $a \\sim b$, \"$a$ is infinitely larger than $b$\" and \"$b$ is infinitely larger than $a$\" holds, so the classes of nonzero elements are totally ordered by size.\nExample: in $\\Z^2$ with the lexicographic order, $(0, 2) \\sim (0, 3)$, since $\\abs{(0, 2)} \\le 1 \\cdot (0, 3)$ and $(0, 3) \\le 2 \\cdot (0, 2)$.\nNon-example: $(1, 0) \\not\\sim (0, 1)$, since $n \\cdot (0, 1) = (0, n) \u003c (1, 0)$ for all $n$, so no multiple of $(0, 1)$ bounds $(1, 0)$.\nSo $\\Z^2$ with this order has exactly two classes of nonzero elements, those with first coordinate $0$ and the others.\nThe group $G$ is Archimedean if and only if all its nonzero elements are equivalent.\nConventions on the direction of the order of classes differ between texts.\n",
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