On the Structure of the Burnside Semigroups

Alair Pereira do Lago

For a fixed set of generators A and for n ≥ 1 and m ≥ 1 integers, the free associative semigroup that satisfies the equation x^{n+m} = x^{n} is called a Burnside semigroup. If we allow n = 0 we have a Burnside group and it can be infinite for m large enough.

We present some structural results on the Burnside semigroups.

In order to classify the structure of these semigroups we have to split them in three cases: n = 1, n = 2 and n ≥ 3. Our main interest here is the case n = 2 whose structure was practically unknown until now. We have a fairly complete set of results for the case n = 2 but here we restrict ourselves to the statement of two facts: one old and one new.

It was known that the semigroups are infinite (with infinitely many J - classes) when n ≥ 2 for more than one generator, while the semigroup is finite for m small enough when n = 1. Anyway, there are only finitely many J -classes for n = 1 and any m.

The maximal subgroups of these semigroups are proved to be free Burnside groups satisfying x^{m} = 1 whose set of generators is associated with a special graph. The cardinality of the generating set is the cyclomatic number of this graph. When n ≥ 3 this special graph is a cycle and the maximal subgroups are cyclic groups of order m. We have examples with an arbitrary number of generators both for n = 1 and for n = 2. Hence we might have infinite maximal subgroups for m large enough. Some of these results in the case n = 1 have been known.