Finiteness Conditions on Skew Braces and Solutions of the Yang–Baxter Equation

A finite non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation gives rise to a structure skew brace $B(X,r)$ that is a $\lambda_f$-skew brace, i.e. every element has finitely many $\lambda$-images, and whose additive group is $FC$. This motivates the study of finiteness conditions on skew braces. We first study the general class of $\lambda_f$ skew braces and the subclass where the additive group is $FC$, showing that these properties share a resemblance to finite conjugacy, having an analog of the $FC$-center and several analogous structural results. Furthermore, by passing through the structure skew brace of a solution, this property measures whether elements are contained in a finite decomposition factor, identifying a class of infinite solutions that may exhibit similar properties to finite ones. Finally, we show that for a sub skew brace where both groups have finite index, both indices need to coincide and that such a sub skew brace contains a strong left ideal of finite index.