Free idempotent generated semigroups and endomorphism monoids of independence algebras
Dandan Yang, Victoria A. R. Gould · Semigroup Forum · 2016
We study maximal subgroups of the free idempotent generated semigroup $${\text {IG}}(E),$$ where E is the biordered set of idempotents of the endomorphism monoid $${\text {End}}\mathbf {A}$$ of an independence algebra $$\mathbf {A}$$ , in the case where $$\mathbf {A}$$ has no constants and has finite rank n. It is shown that when $$n\ge 3$$ the maximal subgroup of $${\text {IG}}(E)$$ containing a rank 1 idempotent $$\varepsilon $$ is isomorphic to the corresponding maximal subgroup of $${\text {End}}\mathbf {A}$$ containing $$\varepsilon $$ . The latter is the group of all unary term operations of $$\mathbf {A}$$ . Note that the class of independence algebras with no constants includes sets, free group acts and affine algebras.