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, as also in a rational monoid. Furthermore they also coincide in any epigroup. 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 Ra and Lb 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 (1961) suggest thinking of this situation in terms of an "egg-box": 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 semigroup, is where each element is in an H-class of its own. The egg-box for this semigroup would contain infinitely many eggs, but all eggs are in the same box 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 combinatorics of semigroups have used Green's relations to help enumerate semigroups with certain properties. 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. (By contrast, there are only five groups of order 8.) Example The full transformation semigroup T3 consists of all functions from the set {1, 2, 3} to itself; there are 27 of these. Write (a b c) for the function that sends 1 to a, 2 to b, and 3 to c. Since T3 contains the identity map, (1 2 3), there is no need to adjoin an identity. The egg-box diagram for T3 has three D-classes. They are also J-classes, because these relations coincide for