# Combinatorial Questions for $S\wr_{n} \mathcal{T}_n$ for a semigroup

Al-aadhami, Asawer (2017) Combinatorial Questions for $S\wr_{n} \mathcal{T}_n$ for a semigroup. PhD thesis, University of York.

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## Abstract

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 $\mathcal{T}_{n}$.

Item Type: Thesis (PhD) The University of York > Mathematics (York) MRS Asawer Al-aadhami 26 Feb 2018 16:00 26 Feb 2018 16:00 http://etheses.whiterose.ac.uk/id/eprint/19494

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