Skeleton Key: Subduction Classes in Finite Transformation Semigroups and Green's Relations
Attila Egri-Nagy, Chrystopher L. Nehaniv
TL;DR
The paper investigates how Green's relations in a finite transformation semigroup relate to a subduction relation on image-sets from a faithful action, introducing the skeleton order as a common framework. It defines ${\mathcal I}(X)$ and the subduction preorder ${\subseteq_S}$, and proves a surjective, order-preserving map $\bar{\lambda}_S$ from ${S^1}/{\mathscr J}$ to ${\mathcal I}(X)/\equiv_S$, with inverse-image classes being unions of ${\mathscr J}$-classes; in the right-regular representation these orders become isomorphisms. The work shows that the skeleton can refine or collapse ${\mathscr J}$-class structure, and that subduction may introduce new relations beyond ${\mathscr J}$-class collapse, including nonlinear skeletons. These results aid in understanding and enumerating possible faithful transformation representations and their decompositions, and the constructions are functorial under surjective morphisms.
Abstract
We establish key connections between Green's $\cal J$- and $\cal L$-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 $\leq_{\cal L}$-order and $\leq_{\cal J}$-order on the semigroup and through the inclusion order on image sets. For right regular representations, the correspondence between the $\cal J$-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 $\cal H$-relation.