A (left) skew brace is a triple \((G, \star, \circ)\), where \((G, \star)\) and \((G, \circ)\) are both groups, and the two operations are related by the identity \(a \circ (b \star c) = (a \circ b) \star a^{-1} \star (a \circ c)\), valid for all \(a, b, c \in G\). Here \(a^{-1}\) denotes the inverse in \((G, \star)\). Equivalently, for each \(a \in G\), the maps \(x\mapsto (a \circ x) \star a^{-1}\) are endomorphisms of \((G, \star)\).
Skew (left) braces can be considered as generalizations of primitive rings, and are widely studied, among others for their connections to Hopf Galois structures.
In the paper under review the author starts building a theory of bi-skew braces; these are the braces \((G, \star, \circ)\) such that \((G, \circ, \star)\) is also a brace. The author provides several methods for constructing such bi-skew braces, one involving radical rings, and the other semidirect products. He also shows how to obtain via bi-skew braces a result of T. Crespo et al. [J. Algebra 457, 312–322 (2016; Zbl 1408.16024)], which states that if \(G\) is a group which is a semidirect product of the groups \(H\) and \(K\), then any Galois extension with Galois group \(G\) admits a Hopf Galois structure of type \(H \times K\).