Abstract
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.
Highlights
Let S be a semigroup with set E = E(S) of idempotents, and let E denote the subsemigroup of S generated by E
Given the common properties shared by full transformation monoids and matrix monoids, Gould [18] and Fountain and Lewin [17] studied the endomorphism monoid End A of an independence algebra A
Other natural question arise: for a particular biordered set E, what are the maximal subgroups of IG(E)? Gray and Ruškuc [22] investigated the maximal subgroups of IG(E), where E is the biordered set of idempotents of a full transformation monoid Tn, 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 Tn containing e, and to the symmetric group Sr
Summary
Gray and Ruškuc [22] investigated the maximal subgroups of IG(E), where E is the biordered set of idempotents of a full transformation monoid Tn, 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 Tn containing e, and to the symmetric group Sr. With the above established, other natural question arise: for a particular biordered set E, what are the maximal subgroups of IG(E)? Results for the biordered sets of idempotents of the full transformation monoid Tn, the matrix monoid Mn(D) of all n × n matrices over a division ring D and the endomorphism monoid End Fn(G) of a free (left) G-act Fn(G) suggest that it may well be worth investigating maximal subgroups of IG(E), where E is the biordered set of idempotents of the endomorphism monoid End A of an independence algebra. Algebra to [28], [6] and [20]
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