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Instead of working directly with a semigroup S, we define Green's relations over the monoid S¹. (S¹ is "S with an identity adjoined if necessary"; if S is not already a monoid, a new element is adjoined and defined to be an identity.) This ensures that principal ideals generated by some semigroup element do indeed contain that element. For an element a of S, the relevant ideals are:

• The principal left ideal generated by a: . This is the same as , which is .
• The principal right ideal generated by a: , or equivalently .
• The principal two-sided ideal generated by a: , or .

1 The L, R, and J relations

For elements a and b of S, Green's relations L, R and J are defined by

• a L b if and only if S¹ a = S¹ b.
• a R b if and only if a S¹ = b S¹.
• a J b if and only if S¹ a S¹ = S¹ b S¹.

That is, a and b are L-related if they generate the same left ideal; R-related if they generate the same right ideal; and J-related if they generate the same two-sided ideal. These are equivalence relations on S, so each of them yields a partition of S into equivalence classes. The L-class of a is denoted L_a (and similarly for the other relations).

Green used the lowercase Fraktur letters , and for these relations, and wrote for a L b (and likewise for R and J). Mathematicians today tend to use script letters such as instead, and replace Green's modular arithmetic-style notation with the infix style used here. Ordinary letters are used for the equivalence classes.

The L and R relations are left-right dual to one another; theorems concerning one can be translated into similar statements about the other. For example, L is right-compatible: if a L b and c is another element of S, then ac L bc. Dually, R is left-compatible: if a R b, then ca R cb.

If S is commutative, then L, R and J coincide.

2 The H and D relations

The remaining relations are derived from L and R. Their intersection is H:

a H b if and only if a L b and a R b.

This is also an equivalence relation on S. An important theorem states that the equivalence class H_e, where e is an idempotent, is a subgroup of S (its identity is e, and all elements have inverses), and indeed is the largest subgroup of S containing e. For example, in the transformation semigroup on n elements, T_n, the H-class of the identity map is the symmetric group S_n.

The class H_a is the intersection of L_a and R_a. More generally, the intersection of any L-class with any R-class is either an H-class or the empty set.

Finally, D is defined by

a D b if and only if there exists a c in S such that a L c and c R b.

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.

There is also a formulation of D in terms of equivalence classes, derived directly from the above definition:

a D b if and only if the intersection of R_a and L_b is not empty.

Consequently, the D-classes of a semigroup can be seen as unions of L-classes, as unions of R-classes, or as unions of H-classes. Clifford and Preston suggest thinking of this situation in terms of an egg carton:

Each row of eggs represents an R-class, and each column an L-class; the eggs themselves are the H-classes. For a group, there is only one egg, because all five of Green's relations coincide, and make all group elements equivalent. The opposite case, found for example in the bicyclic semigroupIn mathematics, the bicyclic semigroup is an algebraic object important for the structure theory of semigroups. Although it is in fact a monoid, it is usually referred to as simply a semigroup. The first published description of this object was given by E, is where each element is in an H-class of its own. The carton for this semigroup would contain infinitely many eggs, but all eggs are in the same carton because there is only one D-class. (A semigroup for which all elements are D-related is called bisimple.)

It can be shown that within a D-class, all H-classes are the same size. For example, the transformation semigroup T₄ contains four D-classes, within which the H-classes have 1, 2, 6, and 24 elements respectively.

Recent advances in the combinatoricsCombinatorics Discrete mathematics Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria, and is in particular concerned with "counting" the objects in those collections enumerative combinatori of semigroups have used Green's relations to help enumerate semigroups with certain properties. A typical result, by Satoh, Yama, and Tokizawa, shows that there are exactly 1,843,120,128 semigroups of order 8, including 221,805 which are commutative; their work is based on a systematic exploration of possible D-classes. (By contrast, there are only five groups of order 8The following list in mathematics contains the finite groups of small order up to group isomorphism. The list can be used to determine which known group a given finite group G is isomorphic to: first determine the order of G then look up the candidates fo.)