 
Summary: NORMAL FORMS FOR FREE APERIODIC SEMIGROUPS
JONATHAN P. MCCAMMOND 1
Abstract. The implicit operation is the unary operation which sends each
element of a finite semigroup to the unique idempotent contained in the sub
semigroup it generates. Using there is a welldefined algebra which is known
as the free aperiodic semigroup. In this article we show that for each n, the n
generated 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 recursive. This
completes a crucial step towards showing that the KrohnRhodes complexity
of every finite semigroup is decidable.
Contents
1. Introduction 1
2. Implicit operations 2
3. Free aperiodic semigroups 4
4. The normal form algorithm in rank 1 6
5. Consequences of the normal form algorithm 10
6. The normal form algorithm in rank i + 1 14
7. The term problem for Fn 20
8. Burnside semigroups 20
