On some Garsideness properties of structure groups of set-theoretic solutions of the Yang-Baxter equation

There exists a multiplicative homomorphism from the braid group B(k+1 )on k + 1 strands to the Temperley-Lieb algebra TLk. Moreover, the homomorphic images in TLk of the simple elements form a basis for the vector space underlying TLk. In analogy with the case of B-k, there exists a multiplicative homomorphism from the structure group G(X, r) of a non-degenerate, involutive set-theoretic solution (X, r), with |X|=n , to an algebra, which extends to a homomorphism of algebras. We construct a finite basis of the underlying vector space of the image of G(X, r) using the Garsideness properties of G(X, r).