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From Relation to Emulation and Interpretation: Implementing the Covering Lemma for Finite Transformation Semigroups

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arXiv:2404.11923v1 Announce Type: new
Abstract: We give a computational 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 recover the information lost in the morphism and package that into a third transformation semigroup in such a way that a cascade product (subsemigroup of the wreath product) can emulate $(X,S)$, providing a hierarchical way of understanding its structure and dynamics: $(X,S)\hookrightarrow (Y,T)\wr (Z,U)$.
The implementation complements the existing tools for the holonomy decomposition algorithm as 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 computational Covering Lemma could be the easiest way of understanding the hierarchical decompositions of transformation semigroups and thus the celebrated Krohn-Rhodes theory.

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