From Relation to Emulation and Interpretation: Computer Algebra Implementation of the Covering Lemma for Finite Transformation Semigroups

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.