  We show the intimate connection between various mathematical notions that are
currently under active investigation: a class of Garside monoids, with a "nice"
Garside element, certain monoids $S$ with quadratic relations, whose monoidal
algebra $A= k[S]$ has a Frobenius Koszul dual $A^{!}$ with regular socle, the
monoids of skew-polynomial type (or equivalently, binomial skew-polynomial
rings) which were introduced and studied by the author and in 1995 provided a
new class of Noetherian Artin-Schelter regular domains, and the square-free
set-theoretic solutions of the Yang-Baxter equation. There is a beautiful
symmetry in these objects due to their nice combinatorial and algebraic
properties.
