Idempotent generation in the endomorphism monoid of a uniform partition

Denote by Tn and Sn the full transformation semigroup and the symmetric group on the set {1, . . . , n}, and En = {1} ∪ (Tn Sn). Let T (X,P) denote the monoid of all transformations of the finite set X preserving a uniform partition P of X into m subsets of size n, where m, n ≥ 2. We enumerate the idempotents of T (X,P), and describe the submonoid S = _E_ generated by the idempotents E = E(T (X,P)). We show that S = S1 ∪ S2, where S1 is a direct product of m copies of En, and S2 is a wreath product of Tn with Tm Sm. 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 En. Note: Some of the scientific symbols cannot be represented correctly in the abstract. Please read with caution and refer to the original publication