Normal Forms for Free Aperiodic Semigroups

The implicit operation ω is the unary operation which sends each element of a finite semigroup to the unique idempotent contained in the subsemigroup it generates. Using ω there is a well-defined algebra which is known as the free aperiodic semigroup. In this article we show that for each n, the ngenerated free aperiodic semigroup is defined by a finite list of pseudoidentities and has a decidable word problem. In the language of implicit operations, this shows that the pseudovariety of finite aperiodic semigroups is κ-recursive. This completes a crucial step towards showing that the Krohn-Rhodes complexity of every finite semigroup is decidable.