Green's relations

Summary

Green's relations are five equivalence relations $\mathcal{L}$, $\mathcal{R}$, $\mathcal{J}$, $\mathcal{H}$ and $\mathcal{D}$ on a semigroup. 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 is a subgroup, by Green's theorem.