Combinatorial Questions for $S\wr_{n} \mathcal{T}_n$ for a semigroup
Asawer Al-Aadhami · White Rose eTheses Online (University of Leeds, The University of Sheffield, University of York) · 2017
We study combinatorial questions for the wreath product $S\\wr_{n}\\mathcal{T}_{n}$ (properly, $S\\wr_{\\underline{n}}\\mathcal{T}_{n}$) and related semigroups, where $S$ is a monoid and $\\mathcal{T}_n$ is the full transformation monoid on $\\underline{n}=\\{ 1,2,\\hdots, n\\}$. It is well known that $S\\wr_{n}\\mathcal{T}_{n}$ is isomorphic to the endomorphism monoid of a free $S$-act $F_{n}(S)$ on $n$ generators and if $S$ is a group, $F_{n}(S)$ is an example of an independence algebra. We determine the number of idempotents of $S\\wr_{n}\\mathcal{T}_{n}$, first in the more straightforward case where $S$ is a group. We investigate the monoid of partial endomorphisms $\\mathcal{PT}_{{\\bf A}}$ of an independence algebra ${\\bf A}$, focussing on the special case where $\\bf A$ is $\\bf{F_{n}(G)}$. We determine Green's relations and Green's pre-orders on $\\mathcal{PT}_{{\\bf F_{n}(G)}}$. We also obtain formulae for the number of idempotents and the number of nilpotents in $\\mathcal{PT}_{{\\bf F_{n}(G)}}$. We specialise Lavers' technique in order to construct a presentation for $M^{n}\\rtimes \\mathcal{T}_{n}$ from presentations of $M^{n}$ and $\\mathcal{T}_{n}$.