Set-theoretic solutions of the Yang-Baxter equation, RC-calculus, and Garside germs

Building on a result by W. Rump, we show how to exploit the right-cyclic law (xy)(xz) = (yx)(yz) in order to investigate the structure groups and monoids attached with (involutive nondegenerate) set-theoretic solutions of the Yang Baxter equation. We develop a sort of right-cyclic calculus, and use it to obtain short proofs for the existence both of the Garside structure and of the I-structure of such groups. We describe finite quotients that play for the considered groups the role that Coxeter groups play for Artin Tits groups. (C) 2015 Elsevier Inc. All rights reserved.