Gröbner–Shirshov basis for the braid group in the Artin–Garside generators

Using [Bokut, L., Fong, Y., Ke, W.-F., Shiao, L-S., 2003. Gröbner–Shirshov basis for the braid semigroup. In: Shum, K.-P. (Ed.), Advances in Algebra and Related Topics. Proceedings of the ICM2002 Satellite Conference on Algebra, Hong Kong. World Scientific, River Edge, pp. 14–25], we find a Gröbner–Shirshov basis S for the braid group Bn+1 in the Artin–Garside generators. We prove that S-irreducible words of the Bn+1 coincide with the Garside normal form words. It gives a new proof of the uniqueness of the Garside normal form of a word, as well as a new proof that the semigroup of positive braids is a subsemigroup into Bn+1.