Left non-degenerate set-theoretic solutions of the Yang-Baxter equation and dynamical extensions of q -cycle sets

A first aim of this paper is to give sufficient conditions on left non-degenerate bijective set-theoretic solutions of the Yang-Baxter equation so that they are non-degenerate. In particular, we extend the results on involutive solutions obtained by Rump in A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation, Adv. Math. 193 (2005) 40-55, https://doi.org/10.1016/j.aim. 2004.03.019 and answer positively a question posed by Cedo et al. in Question 4.2 in Structure monoids of set-theoretic solutions of the Yang Baxter equation, preprint (2019), https://arxiv.org/abs/1912.09710. Moreover, we develop a theory of extensions for left non-degenerate set-theoretic solutions of the Yang-Baxter equation that allows one to construct new families of set-theoretic solutions.