The Word Problem and the Conjugacy Problem for the Structure Skew Brace of a Solution of the Yang–Baxter Equation are Solvable

The aim of this paper is to prove that the Word Problem and the Conjugacy Problem for the structure left skew brace associated with a finite non-degenerate solution of the Yang–Baxter equation are solvable. In order to achieve this result, we need to introduce the concept of (almost) polycyclic left skew brace and to develop a general theory showing that almost polycyclic left skew braces are controlled by their finite homomorphic images. Our results provide us with the first class of infinite solutions of the Yang–Baxter equation on which is possible to work in an algorithmic way: the class of almost polycyclic solutions.