  Denote by $\mathcal T_n$ and $\mathcal S_n$ the full transformation semigroup
and the symmetric group on the set $\{1,\ldots,n\}$, and $\mathcal
E_n=\{1\}\cup(\mathcal T_n\setminus \mathcal S_n)$. Let $\mathcal T(X,\mathcal
P)$ denote the set of all transformations of the finite set $X$ preserving a
uniform partition $\mathcal P$ of $X$ into $m$ subsets of size $n$, where
$m,n\geq2$. We enumerate the idempotents of $\mathcal T(X,\mathcal P)$, and
describe the subsemigroup $S=\langle E\rangle$ generated by the idempotents
$E=E(\mathcal T(X,\mathcal P))$. We show that $S=S_1\cup S_2$, where $S_1$ is a
direct product of $m$ copies of $\mathcal E_n$, and $S_2$ is a wreath product
of $\mathcal T_n$ with $\mathcal T_m\setminus \mathcal S_m$. We calculate the
rank and idempotent rank of $S$, showing that these are equal, and we also
classify and enumerate all the idempotent generating sets of minimal size. In
doing so, we also obtain new results about arbitrary idempotent generating sets
of $\mathcal E_n$.
