The relations are named for James Alexander Green, who introduced them in a paper of 1951.
Instead of working directly with a semigroup S, it is convenient to define Green's relations over the monoid S1.
The L-classes and R-classes can be equivalently understood as the strongly connected components of the left and right Cayley graphs of S1.
instead, and replace Green's modular arithmetic-style notation with the infix style used here.
The L and R relations are left-right dual to one another; theorems concerning one can be translated into similar statements about the other.
More generally, the intersection of any L-class with any R-class is either an H-class or the empty set.
In a monoid M, the class H1 is traditionally called the group of units.
[4] (Beware that unit does not mean identity in this context, i.e. in general there are non-identity elements in H1.
In the language of lattices, D is the join of L and R. (The join for equivalence relations is normally more difficult to define, but is simplified in this case by the fact that a L c and c R b for some c if and only if a R d and d L b for some d.) As D is the smallest equivalence relation containing both L and R, we know that a D b implies a J b—so J contains D. In a finite semigroup, D and J are the same,[5] as also in a rational monoid.
Clifford and Preston (1961) suggest thinking of this situation in terms of an "egg-box":[9] Each row of eggs represents an R-class, and each column an L-class; the eggs themselves are the H-classes.
The opposite case, found for example in the bicyclic semigroup, is where each element is in an H-class of its own.
For example, the transformation semigroup T4 contains four D-classes, within which the H-classes have 1, 2, 6, and 24 elements respectively.
A typical result (Satoh, Yama, and Tokizawa 1994) shows that there are exactly 1,843,120,128 non-equivalent semigroups of order 8, including 221,805 that are commutative; their work is based on a systematic exploration of possible D-classes.
The full transformation semigroup T3 consists of all functions from the set {1, 2, 3} to itself; there are 27 of these.
Following the first route, analogous versions of Green's relations have been defined for semirings (Grillet 1970) and rings (Petro 2002).
For the second kind of generalisation, researchers have concentrated on properties of bijections between L- and R- classes.
These bijections are right and left translations, restricted to the appropriate equivalence classes.
(Conventionally, arguments are written on the right for Λ, and on the left for Ρ.)
There are several choices of partial transformation semigroup that yield the original relations.
They subsume Wallace's work, as well as various other generalised definitions proposed in the mid-1970s.
The full axioms are fairly lengthy to state; informally, the most important requirements are that both Λ and Ρ should contain the identity transformation, and that elements of Λ should commute with elements of Ρ.