  We establish the connection between Green's ${\mathcal J}$-classes and the
subduction equivalence classes defined on the image sets of an action of a
semigroup. The construction of the skeleton order (on subduction equivalence
classes) is shown to depend in a functorial way on transformation semigroups
and surjective morphisms, and to factor through the $\leq_{\mathcal L}$-order
and $\leq_{\mathcal J}$-order on the semigroup and through the inclusion order
on image sets. For right regular representations, the correspondence between
the ${\mathcal J}$-class order and the skeleton is one of isomorphism.
