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

Green's ${\mathcal J}$-classes and Subduction Classes in Finite Transformation Semigroups