# Chinese syzygies by insertions

## TL;DR

This paper develops a finite, convergent algebraic presentation for the Chinese monoid using insertion algorithms, enabling new representations and actions on categories.

## Contribution

It introduces a novel semi-quadratic presentation with column generators and extends it to a coherent presentation based on insertion algorithms.

## Key findings

- Constructed a finite convergent presentation for Chinese monoids.
- Extended the presentation to a coherent one with relations among insertion algorithms.
- Facilitated new categorical representations of Chinese monoids.

## Abstract

We construct a finite convergent semi-quadratic presentation for the Chinese monoid by adding column generators and using combinatorial properties of insertion algorithms on Chinese staircases. We extend this presentation into a coherent one whose generators are columns, rewriting rules are defined by insertion algorithms, and whose syzygies are defined as relations among insertion algorithms. Such a coherent presentation is used for representations of Chinese monoids, in particular, it is a way to describe actions of Chinese monoids on categories.