Chain conditions on skew braces and solutions of the Yang-Baxter Equation

Abstract

Classical works of Hall and McLain show that solubility and local nilpotency play a key role in deriving finite generation in groups from maximal or minimal conditions on normal subgroups. In this work, brace-theoretical analogues of Hall's and McLain's results are analysed for skew braces satisfying the maximal or minimal condition on ideals. We also introduce finiteness and chain conditions on non-degenerate set-theoretic solutions of the Yang-Baxter equation, and their impact on associated structure and permutation skew braces of solutions is also described.