Free idempotent generated semigroups and endomorphism monoids of independence algebras

Abstract

We study maximal subgroups of the free idempotent generated semigroup IG(E), where E is the biordered set of idempotents of the endomorphism monoid End A of an independence algebra A, in the case where A has no constants and has finite rank n. It is shown that when n ≥ 3 the maximal subgroup of IG(E) containing a rank 1 idempotent ε is isomorphic to the corresponding maximal subgroup of End A containing ε. The latter is the group of all unary term operations of A. Note that the class of independence algebras with no constants includes sets, free group acts and affine algebras.

Presentations for singular wreath products

“…The first counter-example was provided in [8] and, soon after, it was shown by Gray and Ruškuc [46] that every group occurs as a maximal subgroup of some FIGS. A recent focus has, therefore, been to describe the maximal subgroups of FIGSs arising from biordered sets of well-known semigroups [11,12,15,18,45,61], while relatively fewer studies of the global structure of a FIGS have been carried out [12,13,19]. The Gray-Ruškuc result [46] has now been proved in a number of different ways [11,20,42], with endomorphism monoids of free G-acts playing a key role in some of these later proofs, and providing a natural biordered set for the construction.…”