Green's relations
In mathematics, Green's relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. The relations are named for James Alexander Green, who introduced them in a paper of 1951. John Mackintosh Howie, a prominent semigroup theorist, described this work as "so allpervading that, on encountering a new semigroup, almost the first question one asks is 'What are the Green relations like?'" (Howie 2002). The relations are useful for understanding the nature of divisibility in a semigroup; they are also valid for groups, but in this case tell us nothing useful, because groups always have divisibility.
Instead of working directly with a semigroup S, it is convenient to define Green's relations over the monoid S^{1}. (S^{1} 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 twosided ideal generated by a: , or .
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 S^{1} a = S^{1} b.
 a R b if and only if a S^{1} = b S^{1}.
 a J b if and only if S^{1} a S^{1} = S^{1} b S^{1}.
That is, a and b are Lrelated if they generate the same left ideal; Rrelated if they generate the same right ideal; and Jrelated if they generate the same twosided ideal. These are equivalence relations on S, so each of them yields a partition of S into equivalence classes. The Lclass of a is denoted L_{a} (and similarly for the other relations).
Green used the lowercase blackletter , 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 arithmeticstyle notation with the infix style used here. Ordinary letters are used for the equivalence classes.
The L and R relations are leftright dual to one another; theorems concerning one can be translated into similar statements about the other. For example, L is rightcompatible: if a L b and c is another element of S, then ac L bc. Dually, R is leftcompatible: if a R b, then ca R cb.
If S is commutative, then L, R and J coincide.
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. The class H_{a} is the intersection of L_{a} and R_{a}. More generally, the intersection of any Lclass with any Rclass is either an Hclass or the empty set.
Green's Theorem states that for any Hclass H of a semigroup S either (i) or (ii) and H is a subgroup of S. An important corollary is 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. No Hclass can contain more than one idempotent, thus H is idempotent separating. In a monoid M, H_{1} is traditionally called the group of units.^{[1]} (Beware that unit does not mean identity in this context, i.e. in general there are nonidentity elements in H_{1}. The "unit" terminology comes from ring theory.) For example, in the transformation monoid on n elements, T_{n}, the group of units is the symmetric group S_{n}.
Finally, D is defined: 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.^{[2]} as also in a rational monoid.^{[3]} Furthermore they also coincide in any epigroup.^{[4]}
There is also a formulation of D in terms of equivalence classes, derived directly from the above definition:^{[5]}
 a D b if and only if the intersection of R_{a} and L_{b} is not empty.
Consequently, the Dclasses of a semigroup can be seen as unions of Lclasses, as unions of Rclasses, or as unions of Hclasses. Clifford and Preston (1961) suggest thinking of this situation in terms of an "eggbox":^{[6]}
Each row of eggs represents an Rclass, and each column an Lclass; the eggs themselves are the Hclasses. 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 Hclass of its own. The eggbox for this semigroup would contain infinitely many eggs, but all eggs are in the same box because there is only one Dclass. (A semigroup for which all elements are Drelated is called bisimple.)
It can be shown that within a Dclass, all Hclasses are the same size. For example, the transformation semigroup T_{4} contains four Dclasses, within which the Hclasses 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 nonequivalent semigroups of order 8, including 221,805 which are commutative; their work is based on a systematic exploration of possible Dclasses. (By contrast, there are only five groups of order 8.)
Example
The full transformation semigroup T_{3} consists of all functions from the set {1, 2, 3} to itself; there are 27 of these. Write (a b c) for the function which sends 1 to a, 2 to b, and 3 to c. Since T_{3} contains the identity map, (1 2 3), there is no need to adjoin an identity.
The eggbox diagram for T_{3} has three Dclasses. They are also Jclasses, because these relations coincide for a finite semigroup.
 
 

In T_{3}, two functions are Lrelated if and only if they have the same image. Such functions appear in the same column of the table above. Likewise, the functions f and g are Rrelated if and only if
 f(x) = f(y) ⇔ g(x) = g(y)
for x and y in {1, 2, 3}; such functions are in the same table row. Consequently, two functions are Drelated if and only if their images are the same size.
The elements in bold are the idempotents. Any Hclass containing one of these is a (maximal) subgroup. In particular, the third Dclass is isomorphic to the symmetric group S_{3}. There are also six subgroups of order 2, and three of order 1 (as well as subgroups of these subgroups). Six elements of T_{3} are not in any subgroup.
Generalisations
There are essentially two ways of generalising an algebraic theory. One is to change its definitions so that it covers more or different objects; the other, more subtle way, is to find some desirable outcome of the theory and consider alternative ways of reaching that conclusion.
Following the first route, analogous versions of Green's relations have been defined for semirings (Grillet 1970) and rings (Petro 2002). Some, but not all, of the properties associated with the relations in semigroups carry over to these cases. Staying within the world of semigroups, Green's relations can be extended to cover relative ideals, which are subsets that are only ideals with respect to a subsemigroup (Wallace 1963).
For the second kind of generalisation, researchers have concentrated on properties of bijections between L and R classes. If x R y, then it is always possible to find bijections between L_{x} and L_{y} that are Rclasspreserving. (That is, if two elements of an Lclass are in the same Rclass, then their images under a bijection will still be in the same Rclass.) The dual statement for x L y also holds. These bijections are right and left translations, restricted to the appropriate equivalence classes. The question that arises is: how else could there be such bijections?
Suppose that Λ and Ρ are semigroups of partial transformations of some semigroup S. Under certain conditions, it can be shown that if x Ρ = y Ρ, with x ρ_{1} = y and y ρ_{2} = x, then the restrictions
 ρ_{1} : Λ x → Λ y
 ρ_{2} : Λ y → Λ x
are mutually inverse bijections. (Conventionally, arguments are written on the right for Λ, and on the left for Ρ.) Then the L and R relations can be defined by
 x L y if and only if Λ x = Λ y
 x R y if and only if x Ρ = y Ρ
and D and H follow as usual. Generalisation of J is not part of this system, as it plays no part in the desired property.
We call (Λ, Ρ) a Green's pair. There are several choices of partial transformation semigroup that yield the original relations. One example would be to take Λ to be the semigroup of all left translations on S^{1}, restricted to S, and Ρ the corresponding semigroup of restricted right translations.
These definitions are due to Clark and Carruth (1980). They subsume Wallace's work, as well as various other generalised definitions proposed in the mid1970s. 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.
See also
References
 ↑ Howie, p. 171
 ↑ Gomes, Pin & Silva (2002), p. 94
 ↑ Sakarovitch, Jacques (September 1987). "Easy multiplications I. The realm of Kleene's theorem". Information and Computation. 74 (3): 173–197. doi:10.1016/08905401(87)900204. Zbl 0642.20043.
 ↑ Peter M. Higgins (1992). Techniques of semigroup theory. Oxford University Press. p. 28. ISBN 9780198535775.
 ↑ Lawson (2004) p.219
 ↑ Lawson (2004) p.220
 Generalized Green's theories, C. E. Clark and J. H. Carruth. Semigroup Forum 20(2) 1980, pp 95–127.
 The algebraic theory of semigroups, A. H. Clifford and G. B. Preston. American Mathematical Society, 1961 (volume 1), 1967 (volume 2). Green's relations are introduced in Chapter 2 of the first volume.
 On the structure of semigroups, J. A. Green. Annals of Mathematics (second series) 54(1), July 1951, pages 163–172.
 Grillet, Mireille P. (1970). "Green's relations in a semiring". Port. Math. 29: 181–195. Zbl 0227.16029.
 An introduction to semigroup theory, J. M. Howie. Academic Press, 1976. ISBN 0123569508. An updated version is available as Fundamentals of semigroup theory, Oxford University Press, 1995. ISBN 0198511949.
 Semigroups, Past, Present and Future, J. M. Howie. Proceedings of the International Conference on Algebra and its Applications, 2002.
 Lawson, Mark V. (2004). Finite automata. Chapman and Hall/CRC. ISBN 1584882557. Zbl 1086.68074.
 Green's relations and minimal quasiideals in rings, Petraq Petro. Comm. Algebra 30(10), 2002, pp 4677–4686.
 Semigroups of order 8, S. Satoh, K. Yama, and M. Tokizawa. Semigroup Forum 49, 1994, pages 7–29.
 Gomes, G.M.S.; Pin, J.E.; Silva, J.E. (2002). Semigroups, algorithms, automata, and languages. Proceedings of workshops held at the International Centre of Mathematics, CIM, Coimbra, Portugal, May, June and July 2001. World Scientific. ISBN 9789812380999. Zbl 1005.00031.