White Rose University Consortium logo
University of Leeds logo University of Sheffield logo York University logo

Free idempotent generated semigroups

YANG, DANDAN (2014) Free idempotent generated semigroups. PhD thesis, University of York.

This is the latest version of this item.

[img] Text (PhD thesis)
final version.pdf
Available under License Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 UK: England & Wales.

Download (890Kb)

Abstract

The study of the free idempotent generated semigroup IG$(E)$ over a biordered set $E$ began with the seminal work of Nambooripad in the 1970s and has seen a recent revival with a number of new approaches, both geometric and combinatorial. Given the universal nature of free idempotent generated semigroups, it is natural to investigate their structure. A popular theme is to investigate the maximal subgroups. It was thought from the 1970s that all such groups would be free, but this conjecture was false. The first example of a non-free group arising in this context appeared in 2009 in an article by Brittenham, Margolis and Meakin. After that, Gray and Ru\v{s}kuc in 2012 showed that {\em any} group occurs as a maximal subgroup of some $\ig(E)$. Following this discovery, another interesting question comes out very naturally: for a particular biordered $E$, which groups can be the maximal subgroups of $\ig(E)$? Several significant results for the biordered sets of idempotents of the full transformation monoid $\mathcal{T}_n$ on $n$ generators and the matrix monoid $M_n(D)$ of all $n\times n$ matrices over a division ring $D,$ have been obtained in recent years, which suggest that it may well be worth investigating maximal subgroups of $\ig(E)$ over the biordered set $E$ of idempotents of the endomorphism monoid of an independence algebra of finite rank $n$. To this end, we investigate another important example of an independence algebra, namely, the free (left) $G$-act $F_n(G)$ of rank $n$, where $n\in \mathbb{N}$, $n\geq 3$ and $G$ is a group. It is known that the endomorphism monoid $\en F_n(G)$ of $F_n(G)$ is isomorphic to a wreath product $G\wr \mathcal{T}_n$. We say that the {\em rank} of an element of $\en F_n(G)$ is the minimal number of (free) generators in its image. Let $E$ be the biordered set of idempotents of $\en F_n(G)$, let $\varepsilon\in E$ be a rank $r$ idempotent, where $1\leq r\leq n.$ For rather straightforward reasons it is known that if $r=n-1$ (respectively, $n$), then the maximal subgroup of $\ig(E)$ containing $\varepsilon$ is free (respectively, trivial). We show, in a transparent way, that, if $r=1$ then the maximal subgroup of IG$(E)$ containing $\varepsilon$ is isomorphic to that of $\en F_n(G)$ and hence to $G$. As a corollary we obtain the 2012 result of Gray and Ru\v{s}kuc that {\em any} group can occur as a maximal subgroup of {\em some} $\ig(E)$. Unlike their proof, ours involves a natural biordered set and very little machinery. However, for higher ranks, a more sophisticated approach is needed, which involves the presentations of maximal subgroups of $\ig(E)$ obtained by Gray and Ru\v{s}kuc, and a presentation of $G\wr\mathcal{S}_r$, where $\mathcal{S}_r$ is the symmetric group on $r$ elements. We show that for $1\leq r\leq n-2$, the maximal subgroup of $\ig(E)$ containing $\varepsilon$ is isomorphic to that of $\en F_n(G)$, and hence to $G\wr\mathcal{S}_r$. By taking $G$ to be trivial, we obtain an alternative proof of the 2012 result of Gray and Ru\v{s}kuc for the biordered set of idempotents of $\mathcal{T}_n.$ After that, we focus on the maximal subgroups of $\ig(E)$ containing a rank 1 idempotent $\varepsilon\in E$, where $E$ is the biordered set of idempotents of the endomorphism monoid $\en \mathbf{A}$ of an independence algebra $\mathbf{A}$ of rank $n$ with no constants, where $n\in \mathbb{N}$ and $n\geq 3.$ It is proved that the maximal subgroup of $\ig(E)$ containing $\varepsilon$ is isomorphic to that of $\en \mathbf{A},$ the latter being the group of all unary term operations of $\mathbf{A}.$ Whereas much of the former work in the literature of $\ig(E)$ has focused on maximal subgroups, in this thesis we also study the general structure of the free idempotent generated semigroup $\ig(B)$ over an arbitrary band $B$. We show that $\ig(B)$ is {\it always} a weakly abundant semigroup with the congruence condition, but not necessarily abundant. We then prove that if $B$ is a quasi-zero band or a normal band for which $\ig(B)$ satisfying Condition $(P)$, then $\ig(B)$ is an abundant semigroup. In consequence, if $Y$ is a semilattice, then $\ig(Y)$ is adequate, that is, it belongs to the quasivariety of unary semigroups introduced by Fountain over 30 years ago. Further, the word problem of $\ig(B)$ is solvable if $B$ is quasi-zero. We also construct a 10-element normal band $B$ for which $\ig(B)$ is not abundant.

Item Type: Thesis (PhD)
Keywords: biordered sets, free G-acts, bands
Academic Units: The University of York > Mathematics (York)
Depositing User: Miss DANDAN YANG
Date Deposited: 19 May 2014 12:32
Last Modified: 19 May 2014 12:32
URI: http://etheses.whiterose.ac.uk/id/eprint/5948

Available Versions of this Item

  • Free idempotent generated semigroups. (deposited 19 May 2014 12:32) [Currently Displayed]

Actions (repository staff only: login required)