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There exists a multiplicative homomorphism from the braid groupon  k  + 1 strands to the Temperley–Lieb algebra  TL k. Moreover, the homomorphic images in  TL kof the simple elements form a basis for the vector space underlying  TL k. 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, 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 ).

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