Idempotent generation in the endomorphism monoid of a uniform partition

Abstract

Denote by ${\mathcal{T}}_n$ and ${\mathcal{S}}_n$ the full transformation semigroup and the symmetric group on the set $\{1,\dots ,n\}$, 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 $m,n\ge 2$. We enumerate the idempotents of $\mathcal{T}(X,\mathcal{P})$, and describe the subsemigroup $S=⟨E⟩$ 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 ${\mathcal{E}}_n$.

17 pages, 6 figure, 6 tables - v2 includes some minor corrections and simplifications suggested by referee - to appear in Comm Alg

References in corpus (3)

• Enumeration of idempotents in diagram semigroups and algebras
• The rank of the semigroup of transformations stabilising a partition of a finite set
• Idempotent rank in the endomorphism monoid of a non-uniform partition

Cited by in corpus (5)

• Motzkin monoids and partial Brauer monoids
• On certain Semigroups of Transformations that preserve a partition
• Regular, Unit-regular, and Idempotent elements of semigroups of transformations that preserve a partition
• On unit-regular elements in various monoids of transformations
• Presentations for singular wreath products