Idempotent Generation in the Endomorphism Monoid of a Uniform Partition

Abstract

© 2016, Copyright © Taylor & Francis Group, LLC. 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.