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  "body": "\n## Summary\n\nGreen's relations are five [[equivalence relations|0G8VEZ3WTK]] $\\mathcal{L}$, $\\mathcal{R}$, $\\mathcal{J}$, $\\mathcal{H}$ and $\\mathcal{D}$ on a [[semigroup|0XMEM406JM]].\nTwo elements are related when they generate the same left, right or two-sided ideal.\nThey sort the elements of a semigroup into classes that play the role of the subgroups and cosets of group theory.\nIn a group, any two elements are related under each of the five relations.\nSo the relations carry information only for semigroups that are not groups.\nAn $\\mathcal{H}$-class that contains an [[idempotent|0N5NJZKKN3]] is a subgroup, by Green's theorem.\n\n## Explanations\n\n### L1 human\n\nIn a group, every element can be reached from every other by multiplying on the left.\nIn a semigroup this fails, and Green's relations record which elements can reach each other.\nTwo elements are $\\mathcal{L}$-related if each is a left multiple of the other, and $\\mathcal{R}$-related if each is a right multiple of the other.\nFor maps of a finite set, two maps are $\\mathcal{R}$-related exactly when they have the same image.\n",
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