Is there a synthetic definition of the $0$-Hecke monoid of $S_n$?

Monoid H_0(S_n) can be defined as a monoid of permutation matrices under a special kind of multiplication, or equivalently of special integer matrices (unit-Monge matrices) under standard tropical multiplication. See my paper for details: https://doi.org/10.1007/s00453-013-9830-z

I am curious about your remark that the monoid's properties are popular as exercises: do you have any examples of those?