Skeleton Key: Subduction Classes in Finite Transformation Semigroups and Green's Relations

We establish key connections between Green’s $\mathcal{J}$- and $\mathcal{L}$-relations on a finite semigroup and the subduction relation defined on the image sets of an action of the same semigroup when it acts faithfully on a finite set. The construction of the skeleton order, the partial order on equivalence classes of the subduction relation, is shown to depend in a functorial way on transformation semigroups and surjective morphisms, and to factor through the Green’s ${\le }_{\mathcal{L}}$-order and ${\le }_{\mathcal{J}}$-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 order is one of isomorphism. Finally, we characterize the relationship between natural subsystems of a transformation semigroup, permutator groups and the $\mathcal{H}$-relation.