Idempotent Generation in the Endomorphism Monoid of a Uniform Partition

Abstract

Denote by 𝒯_n and 𝒮_n the full transformation semigroup and the symmetric group on the set {1,…, n}, and ℰ_n = {1} ∪ (𝒯_n∖𝒮_n). 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 ℰ_n, and S₂ is a wreath product of 𝒯_n with 𝒯_m∖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 ℰ_n.

Key Words:

• Generators
• Idempotents
• Idempotent rank
• Rank
• Transformation semigroups

2010 Mathematics Subject Classification:

• 20M20
• 20M17

ACKNOWLEDGMENT

The authors thank the referee for their careful reading and helpful comments that led to increased clarity and truth.

Notes

¹These sets form the so-called 𝒟-classes of 𝒯_n. No knowledge of Green's relations, which include the 𝒟 relation, will be assumed but the reader may refer to a monograph such as [16, 19] for details if they wish.