Braces and symmetric groups with special conditions

We study left braces satisfying special conditions, or identities. We are particularly interested in the impact of conditions like Raut and lri on the properties of the left brace and its associated solution of the Yang-Baxter equation (YBE). We show that the solution (G,rG) of the YBE associated to the structure group G=G(X,r) (with the natural structure of a left brace) of a nontrivial solution (X,r) of the YBE has multipermutation level 2 if and only if G satisfies lri. It is known that every (left) brace with lri satisfies condition Raut. We prove that for a graded Jacobson radical ring with no elements of additive order two the conditions lri and Raut are equivalent. We construct a finite two-sided brace with condition Raut which does not satisfy lri. We show that a finitely generated two-sided brace which satisfies lri has a finite multipermutation level which is bounded by the number of its generators.