Subgroups of free idempotent generated semigroups need not be free

Abstract

We study the maximal subgroups of free idempotent generated semigroups on a biordered set by topological methods. These subgroups are realized as the fundamental groups of a number of 2-complexes naturally associated to the biorder structure of the set of idempotents. We use this to construct the first example of a free idempotent generated semigroup containing a non-free subgroup.

Free idempotent generated semigroups and endomorphism monoids of independence algebras

“…1 Clearly, IG(E) is idempotent generated, and there is a natural map φ : IG(E) → S, given byēφ = e, such that im φ = S = E . Further, we have the following result, taken from [2,13,15,21,29], which exhibits the close relation between the structure of the regular D-classes of IG(E) and those of S.…”

“…Several papers [27], [30], [33] and [32] established various sufficient conditions guaranteeing that all maximal subgroups are free. However, in 2009, Brittenham, Margolis and Meakin [3] disproved this conjecture by showing that the free abelian group Z ⊕ Z occurs as a maximal subgroup of some IG(E). An unpublished counterexample of McElwee from the 2010s was announced by Easdown [12] in 2011.…”

“…An unpublished counterexample of McElwee from the 2010s was announced by Easdown [12] in 2011. Motivated by the significant discovery in [3], Gray and Ruškuc [21] showed that any group occurs as a maximal subgroup of some IG(E). Their approach is to use Ruškuc's pre-existing machinery for constructing presentations of maximal subgroups of semigroups given by a presentation, and refine this to give presentations of IG(E), and then, given a group G, to carefully choose a biordered set E. Their techniques are significant and powerful, and have other consequences.…”

“…Gray and Ruškuc [22] investigated the maximal subgroups of IG(E), where E is the biordered set of idempotents of a full transformation monoid T n , showing that for any e ∈ E with rank r , where 1 ≤ r ≤ n − 2, the maximal subgroup of IG(E) containing e is isomorphic to the maximal subgroup of T n containing e, and hence to the symmetric group S r . Another strand of this popular theme is to consider the biordered set E of idempotents of the matrix monoid M n (D) of all n × n matrices over a division ring D. By using similar topological methods to those of [3], Brittenham, Margolis and Meakin [2] proved that if e ∈ E is a rank 1 idempotent, then the maximal subgroup of IG(E) containing e is isomorphic to that of M n (D), that is, to the multiplicative group D * of D. Dolinka and Gray [10] went on to generalise the result of [2] to e ∈ E with higher rank r , where r < n/3, showing that the maximal subgroup of IG(E) containing e is isomorphic to the maximal subgroup of M n (D) containing e, and hence to the r dimensional general linear group G L r (D). So far, the structure of maximal subgroups of IG(E) containing e ∈ E, where rank e = r and r ≥ n/3 remains unknown.…”

“…It is known from [3] that every singular square in E is a rectangular band, however the converse is not always true. We say that D is singularisable if an E-square of elements from D is singular if and only if it is a rectangular band.…”