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, using 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.