A Study on Cayley Digraphs of the Full Transformation Semigroup Withrespect to Green's D-Classes

Let Tₙ be the full transformation semigroup on an n-element set and let Dᵣ be the Green D-class of all transformations of rank r. We investigate the looped simple Cayley digraph Γₙ,ᵣ = Cay (Tₙ, Dᵣ) under right multiplication. The key difficulty is that the standard kernel-refinement criterion only guarantees a factorization β = αγ with an unrestricted factor γ; it does not determine when the factor can be chosen with prescribed rank r. We solve this extension problem exactly and prove that α → β if and only if ker(α) refines ker(β) and rank(β) ≤ r ≤ rank(β) + n − rank(α). From this criterion we obtain closed out-degree formulae and a new in-degree formula governed by the block-size profile of ker(β). We identify a source-core decomposition in which all vertices of rank greater than r have in-degree zero and every arc enters the ideal of transformations of rank at most r. We classify all strongly connected components, determine the condensation digraph and its exact height, and derive the exact directed distance from an arbitrary transformation to the constant maps. Weak connectivity, local cyclicity and isomorphism rigidity are then obtained as consequences.