Green's Relations - Generalisations

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 R-class-preserving. (That is, if two elements of an L-class are in the same R-class, then their images under a bijection will still be in the same R-class.) 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 ρ₁ = y and y ρ₂ = x, then the restrictions

ρ₁ : Λ x → Λ y

ρ₂ : 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 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.