  We give a practical computer algebra implementation of the Covering Lemma for
finite transformation semigroups. The lemma states that given a surjective
relational morphism $(X,S)\twoheadrightarrow(Y,T)$, we can establish emulation
by a cascade product (subsemigroup of the wreath product):
$(X,S)\hookrightarrow (Y,T)\wr (Z,U)$. The dependent component $(Z,U)$ contains
the kernel of the morphism, the information lost in the map.
  The implementation complements the existing tools for the holonomy
decomposition algorithm. It gives an incremental method to get a coarser
decomposition when computing the complete skeleton for holonomy is not
feasible. Here, we describe a simplified and generalized algorithm for the
lemma and compare it to the holonomy method. Incidentally, the kernel-based
method could be the easiest way of understanding the hierarchical
decompositions of transformation semigroups and thus the celebrated
Krohn-Rhodes theory.
