Abstract.  It is well-known that certain matrix solutions of the braid or Yang-Baxter equations lead to braided categories, knot invariants, quantum groups and other important constructions. However, these equations are also very interesting at the level of set maps r : X x X --> X x X, where X is a set and r is a bijection, a line of study proposed by Drinfeld. Solutions extend linearly to very special linear solutions but also lead to a great deal of combinatorics, and to algebras with very nice algebraic and homological properties including those relating to Artin-Schelter regularity, Koszulity, being Noetherian domain, the existence of noncommutative Groebner bases.
We extend our recent work on set-theoretic solutions of the Yang-Baxter or braid relations with new results about their automorphism groups, strong twisted unions of solutions and multipermutation solutions. We introduce and study graphs of solutions and use our graphical methods for the computation of solutions of finite order and their automorphisms. Results include a detailed study of solutions of multipermutation level 2.