@slava if you are interested in presentations and monoids, you might be interested that I finally got around to writing down and starting to develop my philosophy of math education.
I selected the Stern-Brocot tree, the Symmetry Group of the Square, Pascal's Triangle, and computer programming as starting points for a study plan based around iterative deepening depth-first search.
Then I accidentally realized combining the Stern-Brocot tree and the Symmetry Group of the Square results in the general modular group GL(2,Z), a fact from which you can derive presentations of GL(2,Z).
It turns out that the Stern-Brocot tree is isomorphic to the free monoid SL(2,N) given by the presentation <L, R>, and this is a submonoid of GL(2,Z), SL(2,Z), PGL(2,Z), and PSL(2,Z).
https://github.com/constructive-symmetry/constructive-symmetry/blob/master/T002_Tools_of_Math_Construction/Part03_Aggregate_Theory.md