  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.
