Quadratic normalisation in monoids

TL;DR

This paper investigates quadratic normalisation processes in monoids, introducing a class parameter to analyze their complexity, and characterizes specific normalisation behaviors, including those from Garside families.

Contribution

It introduces a class parameter to measure normalisation complexity and fully axiomatizes normalisations of class (4, 3), linking them to Garside structures.

Findings

Normalisation of class (4, 3) is fully axiomatized.

Convergence of associated rewriting systems is established.

Characterization of normalisations deriving from Garside families.

Abstract

In the general context of presentations of monoids, we study normalisation processes that are determined by their restriction to length-two words. Garside's greedy normal forms and quadratic convergent rewriting systems, in particular those associated with the plactic monoids, are typical examples. Having introduced a parameter, called the class and measuring the complexity of the normalisation of length-three words, we analyse the normalisation of longer words and describe a number of possible behaviours. We fully axiomatise normalisations of class (4, 3), show the convergence of the associated rewriting systems, and characterise those deriving from a Garside family.