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  "body": "\n## Summary\n\nAn ordered group, or partially ordered group, is a [[group|G5S1M9]] with a [[partial order|0QV2JS9RJ3]] that is preserved by multiplication on both sides.\nThe order is then determined by its positive cone, the set of elements that are at least the identity.\nOrdered groups include the additive groups of ordered fields, groups of functions with the pointwise order, and lattice-ordered groups such as groups of real functions.\nThey are the common setting of the theory of ordered algebraic structures.\n\n## Explanations\n\n### L1 human\n\nIn the integers, if $a \\le b$ then $a + c \\le b + c$: adding the same number to both sides keeps an inequality.\nAn ordered group is a group with an order that has this property.\nThe order does not need to compare every two elements.\nFor example, in the plane one can say $(a, b) \\le (c, d)$ when $a \\le c$ and $b \\le d$.\nThen $(1, 0)$ and $(0, 1)$ are not comparable, but adding a vector to both sides still keeps every inequality.\n",
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  "name": "Ordered group",
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