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 $T_{1}T_{2}$. 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.