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NORMAL FORMS FOR FREE APERIODIC SEMIGROUPS JONATHAN P. MCCAMMOND 1
 

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 well-defined 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 Krohn-Rhodes 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

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics