→ 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 S1, 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 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. See also Schutzenberger group References C. E. Clark and J. H. Carruth (1980) Generalized Green's theories, Semigroup Forum 20(2); 95–127. A. H. Clifford and G. B. Preston (1961) The Algebraic Theory of Semigroups, volume 1, (1967) volume 2, American Mathematical Society, Green's relations are introduced in Chapter 2 of the first volume. J. A. Green (July 1951) "On the structure of semigroups", Annals of Mathematics (second series) 54(1): 163–172. John M. Howie (1976) An introduction to Semigroup Theory, Academic Press . An updated version is available as Fundamentals of Semigroup Theory, Oxford University Press, 1995. . John M. Howie (2002) "Semigroups, Past, Present and Future", Proceedings of the International Conference on Algebra and its Applications, Chulalongkorn University, Thailand Petraq Petro (2002) Green's relations and minimal quasi-ideals in rings, Communications in Algebra 30(10): 4677–4686. S. Satoh, K. Yama, and M. Tokizawa (1994) "Semigroups of order 8", Semigroup Forum 49: 7–29. Semigroup theory