Green's ${\mathcal J}$-classes and Subduction Classes in Finite   Transformation Semigroups
  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. 
Attila Egri-Nagy and Chrystopher L. Nehaniv
math.GR