Abstract
We calculate the rank and idempotent rank of the semigroup ${\mathcal{E}}(X,{\mathcal{P}})$ generated by the idempotents of the semigroup ${\mathcal{T}}(X,{\mathcal{P}})$ which consists of all transformations of the finite set $X$ preserving a nonuniform partition ${\mathcal{P}}$. We also classify and enumerate the idempotent generating sets of minimal possible size. This extends results of the first two authors in the uniform case.
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