We prove that the pseudovariety of monoids of Krohn-Rhodes complexity at most $n$ is not finitely based for all $n\>0$. More specifically, for each pair of positive integers $n,k$, we construct a monoid of complexity $n+1$, all of whose $k$-generated submonoids have complexity at most $n$.

We prove that the pseudovariety of monoids of Krohn-Rhodes complexity at most n is not finitely based for all n>0. More specifically, for each pair of positive integers n,k, we construct a monoid of complexity n+1, all of whose k-generated submonoids have complexity at most n.