---
tag: RFKHKD4KXY
kind: definition
name: Green's relations
msc: [20M10]
defines: S7WJW2YES2
uses: [0XMEM406JM, 0N6R902FG1, 0G8VEZ3WTK, 0N5NJZKKN3, 7ZB7MRKPQ9]
sources:
  - url: https://en.wikipedia.org/wiki/Green's_relations
---

## Definition

Let $S$ be a [[semigroup|0XMEM406JM]].
Let $S^1 = S$ if $S$ is a [[monoid|0N6R902FG1]], and otherwise let $S^1$ be $S$ with an identity element $1$ adjoined.
For $a, b \in S$, *Green's relations* are:

1. $a \mathrel{\mathcal{L}} b$ if $S^1 a = S^1 b$.
2. $a \mathrel{\mathcal{R}} b$ if $a S^1 = b S^1$.
3. $a \mathrel{\mathcal{J}} b$ if $S^1 a S^1 = S^1 b S^1$.
4. $a \mathrel{\mathcal{H}} b$ if $a \mathrel{\mathcal{L}} b$ and $a \mathrel{\mathcal{R}} b$.
5. $a \mathrel{\mathcal{D}} b$ if there is $c \in S$ with $a \mathrel{\mathcal{L}} c$ and $c \mathrel{\mathcal{R}} b$.

## Remarks

All five are [[equivalence relations|0G8VEZ3WTK]]; for $\mathcal{D}$ this uses the fact that $\mathcal{L}$ and $\mathcal{R}$ commute as relations.
Equivalently, $a \mathrel{\mathcal{L}} b$ if and only if $a = xb$ and $b = ya$ for some $x, y \in S^1$.
Always $\mathcal{H} \subseteq \mathcal{L}, \mathcal{R} \subseteq \mathcal{D} \subseteq \mathcal{J}$, and $\mathcal{D} = \mathcal{J}$ when $S$ is finite.

Example: let $T_X$ be the monoid of all maps of a finite set $X$ to itself, with $(fg)(x) = f(g(x))$.
If $f = gh$, then the image of $f$ is contained in the image of $g$.
Conversely, if the images are equal, choose $h(x)$ to be a preimage of $f(x)$ under $g$; then $f = gh$.
So $f \mathrel{\mathcal{R}} g$ if and only if $f$ and $g$ have the same image.
Similarly, $f \mathrel{\mathcal{L}} g$ 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.

Non-example: in the monoid $(\N, +)$, the left ideal generated by $a$ is $a + \N$, so $a \mathrel{\mathcal{L}} b$ only when $a = b$.
Unlike in a group, the identity $0$ and the element $1$ are not $\mathcal{L}$-related.