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  "body": "\n## Definition\n\nLet $S$ be a [[semigroup|0XMEM406JM]].\nLet $S^1 = S$ if $S$ is a [[monoid|0N6R902FG1]], and otherwise let $S^1$ be $S$ with an identity element $1$ adjoined.\nFor $a, b \\in S$, *Green's relations* are:\n\n1. $a \\mathrel{\\mathcal{L}} b$ if $S^1 a = S^1 b$.\n2. $a \\mathrel{\\mathcal{R}} b$ if $a S^1 = b S^1$.\n3. $a \\mathrel{\\mathcal{J}} b$ if $S^1 a S^1 = S^1 b S^1$.\n4. $a \\mathrel{\\mathcal{H}} b$ if $a \\mathrel{\\mathcal{L}} b$ and $a \\mathrel{\\mathcal{R}} b$.\n5. $a \\mathrel{\\mathcal{D}} b$ if there is $c \\in S$ with $a \\mathrel{\\mathcal{L}} c$ and $c \\mathrel{\\mathcal{R}} b$.\n\n## Remarks\n\nAll five are [[equivalence relations|0G8VEZ3WTK]]; for $\\mathcal{D}$ this uses the fact that $\\mathcal{L}$ and $\\mathcal{R}$ commute as relations.\nEquivalently, $a \\mathrel{\\mathcal{L}} b$ if and only if $a = xb$ and $b = ya$ for some $x, y \\in S^1$.\nAlways $\\mathcal{H} \\subseteq \\mathcal{L}, \\mathcal{R} \\subseteq \\mathcal{D} \\subseteq \\mathcal{J}$, and $\\mathcal{D} = \\mathcal{J}$ when $S$ is finite.\n\nExample: let $T_X$ be the monoid of all maps of a finite set $X$ to itself, with $(fg)(x) = f(g(x))$.\nIf $f = gh$, then the image of $f$ is contained in the image of $g$.\nConversely, if the images are equal, choose $h(x)$ to be a preimage of $f(x)$ under $g$; then $f = gh$.\nSo $f \\mathrel{\\mathcal{R}} g$ if and only if $f$ and $g$ have the same image.\nSimilarly, $f \\mathrel{\\mathcal{L}} g$ if and only if $f$ and $g$ have the same fibres, and $f \\mathrel{\\mathcal{D}} g$ if and only if their images have the same size.\n\nNon-example: in the monoid $(\\N, +)$, the left ideal generated by $a$ is $a + \\N$, so $a \\mathrel{\\mathcal{L}} b$ only when $a = b$.\nUnlike in a group, the identity $0$ and the element $1$ are not $\\mathcal{L}$-related.\n",
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