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  "body": "\n## Summary\n\nA linearly ordered group is an [[ordered group|KRDR04RQ7N]] whose order is a [[total order|0FRXBVVR03]].\nAny two of its elements are comparable, and the order is preserved by multiplication on both sides.\nThe additive groups of $\\Z$, $\\Q$ and $\\R$ are the basic examples.\nLinearly ordered abelian groups are the value groups of valuations, and they are described by Hölder's theorem and the Hahn embedding theorem.\n\n## Explanations\n\n### L1 human\n\nA linearly ordered group is a group whose elements stand in one line, from smaller to larger.\nMultiplying by a fixed element on either side shifts the whole line without changing the order of its points.\nThe integers under addition are an example: adding $5$ to every integer keeps them in the same order.\nThe pairs of integers with the dictionary order are another example, where $(0, 1000) \u003c (1, 0)$.\n",
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