Green defined a bunch of relations on any semigroup S which provide a basic structure theory.

Note we write S¹ for S with a 1 adjoined if necessary, which means we can talk about e.g. S¹_x_ as the left ideal generated by x.

Define the L and R relations by:

• x L y <=> S¹x = S¹y
• x R y <=> xS¹ = yS¹

Then L and R are actually commuting congruences because (...), so we have a new relation D = L v R = LR. Also there is a relation H = L ^ R, and another one, J containing all the previous relations:

• x J y <=> S¹xS¹ = S¹yS¹

We have H <= L, R <= D <= J.

The Green theory then explicates this:

• Let X be an H-class. Then either XX ⟂ X, or else XX = X and X contains a unique idempotent and is a group.
• In a finite semigroup, D = J.
• In each D-class, the H-classes all have the same size and can be arranged in a table with L rows and R columns ( Add proofs and links )

This permits a structural decomposition of S into an 'eggbox diagram'.

All of this is illustrated quite nicely by the Green-Rees theory of idempotent semigroups ( link )