Knuth's Coherent Presentations of Plactic Monoids of Type A

TL;DR

This paper constructs finite coherent presentations of plactic monoids of type A using rewriting methods, enabling better understanding of monoid actions on categories and advancing categorical computations.

Contribution

It introduces a finite coherent presentation of plactic monoids of type A based on Squier's rewriting method, extending the known column presentation.

Findings

Finite coherent presentation derived from column presentation.

Application of Squier's rewriting method for coherence.

Reduction to Tietze equivalent presentation with Knuth's generators.

Abstract

We construct finite coherent presentations of plactic monoids of type A. Such coherent presentations express a system of generators and relations for the monoid extended in a coherent way to give a family of generators of the relations amongst the relations. Such extended presentations are used for representations of monoids, in particular, it is a way to describe actions of monoids on categories. Moreover, a coherent presentation provides the first step in the computation of a categorical cofibrant replacement of a monoid. Our construction is based on a rewriting method introduced by Squier that computes a coherent presentation from a convergent one. We compute a finite coherent presentation of a plactic monoid from its column presentation that is known to be finite and convergent. Finally, we show how to reduce this coherent presentation to a Tietze equivalent one having Knuth's generators.