Merge Decompositions, Two-sided Krohn–Rhodes, and Aperiodic Pointlikes

This paper provides short proofs of two fundamental theorems of finite semigroup theory whose previous proofs were significantly longer, namely the two-sided Krohn-Rhodes decomposition theorem and Henckell's aperiodic pointlike theorem. We use a new algebraic technique that we call the merge decomposition. A prototypical application of this technique decomposes a semigroup T into a two-sided semidirect product whose components are built from two subsemigroups T-1, T-2, which together generate T, and the subsemigroup generated by their setwise product T1T2. In this sense we decompose T by merging the subsemigroups T-1 and T-2. More generally, our technique merges semigroup homomorphisms from free semigroups.