---
tag: S7WJW2YES2
kind: concept
name: Green's relations
msc: [20M10]
sources:
  - url: https://en.wikipedia.org/wiki/Green's_relations
same_as:
  wikidata: Q3077964
  wikipedia: Green's relations
---

## Summary

Green's relations are five [[equivalence relations|0G8VEZ3WTK]] $\mathcal{L}$, $\mathcal{R}$, $\mathcal{J}$, $\mathcal{H}$ and $\mathcal{D}$ on a [[semigroup|0XMEM406JM]].
Two elements are related when they generate the same left, right or two-sided ideal.
They sort the elements of a semigroup into classes that play the role of the subgroups and cosets of group theory.
In a group, any two elements are related under each of the five relations.
So the relations carry information only for semigroups that are not groups.
An $\mathcal{H}$-class that contains an [[idempotent|0N5NJZKKN3]] is a subgroup, by Green's theorem.

## Explanations

### L1 human

In a group, every element can be reached from every other by multiplying on the left.
In a semigroup this fails, and Green's relations record which elements can reach each other.
Two 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.
For maps of a finite set, two maps are $\mathcal{R}$-related exactly when they have the same image.