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

Abstract Building on a result by W. Rump, we show how to exploit the right-cyclic law ( x y ) ( x z ) = ( y x ) ( y z ) 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.

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