Presentation of the Motzkin Monoid

TL;DR

This paper studies the algebraic structure of the Motzkin monoid, providing presentations by generators and relations for it and related monoids, and analyzing their decomposition algorithms.

Contribution

It introduces algebraic presentations for the Motzkin monoid and its submonoids, extending diagrammatic decomposition algorithms into an algebraic framework.

Findings

Presented generators and relations for the Right and Left Planar Rook monoids.

Extended Halverson's decomposition algorithm algebraically for the Motzkin monoid.

Used counting arguments to prove sufficiency of the relations.

Abstract

In 2010, Tom Halverson and Georgia Benkart introduced the Motzkin algebra, a generalization of the Temperley-Lieb algebra, whose elements are diagrams that can be multiplied by stacking one on top of the other. Halverson and Benkart gave a diagrammatic algorithm for decomposing any Motzkin diagram into diagrams of three subalgebras: the Right Planar Rook algebra, the Temperley-Lieb algebra, and the Left Planar Rook algebra. We first explore the Right and Left Planar Rook monoids, by finding presentations for these monoids by generators and relations, using a counting argument to prove that our relations suffice. We then turn to the newly-developed Motzkin monoid, where we describe Halverson's decomposition algorithm algebraically, find a presentation by generators and relations, and use a counting argument but with a much more sophisticated algorithm.