Skew-braces and ????-braces

Abstract Skew-braces are ring-like objects arising in connection with Hopf–Galois theory and set-theoretic solutions ???? to the Yang–Baxter equation. Interactions between skew-braces are often related to ????-braces. For example, every ????-brace ???? is given by a pair of skew-braces which induces a ℤ-indexed sequence of skew-braces. The sequence collapses if ???? itself is a skew-brace. The free group over the underlying set of a solution ???? is a ????-brace. Bi-crossed products of skew-braces are shown to be ????-braces, and criteria are developed when they are skew-braces. Two classes of skew-braces are put into a bijective correspondence with special ????-braces, which themselves need not be skew-braces. Characterizations of ????-braces, including one in terms of a graph and one as special modules over skew-braces, are given.