a finite semigroup. In T3, two functions are L-related 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 R-related 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 D-related if and only if their images are the same size. The elements in bold are the idempotents. Any H-class containing one of these is a (maximal) subgroup. In particular, the third D-class is isomorphic to the symmetric group S3. There are also six subgroups of order 2, and three of order 1 (as well as subgroups of these subgroups). Six elements of T3 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 Lx and Ly 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