Free idempotent generated semigroups

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.