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This paper provides short proofs of two fundamental theorems of finitesemigroup theory whose previous proofs were significantly longer, namely thetwo-sided Krohn-Rhodes decomposition theorem and Henckell's aperiodic pointliketheorem, using a new algebraic technique that we call the merge decomposition.A prototypical application of this technique decomposes a semigroup $T$ into atwo-sided semidirect product whose components are built from two subsemigroups$T_1,T_2$, which together generate $T$, and the subsemigroup generated by theirsetwise product $T_1T_2$. In this sense we decompose $T$ by merging thesubsemigroups $T_1$ and $T_2$. More generally, our technique merges semigrouphomomorphisms from free semigroups.

Merge decompositions, two-sided Krohn-Rhodes, and aperiodic pointlikes