  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 $T_1T_2$. 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.
