McCammond's normal forms for free aperiodic semigroups revisited

TL;DR

This paper simplifies the proof of McCammond's algorithm for solving the word problem in finite aperiodic semigroups, using properties of regular languages, and explores new applications of these results.

Contribution

It provides a new, simpler correctness proof for McCammond's algorithm based on regular language properties, enhancing understanding and applications.

Findings

Simplified proof of McCammond's algorithm correctness

New applications derived from the regular language approach

Enhanced understanding of normal forms in aperiodic semigroups

Abstract

This paper revisits the solution of the word problem for $\omega$-terms interpreted over finite aperiodic semigroups, obtained by J. McCammond. The original proof of correctness of McCammond's algorithm, based on normal forms for such terms, uses McCammond's solution of the word problem for certain Burnside semigroups. In this paper, we establish a new, simpler, correctness proof of McCammond's algorithm, based on properties of certain regular languages associated with the normal forms. This method leads to new applications.