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 λf-skew brace, i.e. every element has finitely many λ-images, and whose additive group is FC. This motivates the study of finiteness conditions on skew braces. We first study the general class of λf 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.