Hopf $\pi $-braces and related algebraic structures

In this paper, (weak) Hopf $\pi $-braces are introduced, which can be seen as an extension of Hopf braces and Hopf $\pi $-algebras. A construction of Hopf $\pi $-braces from the Radford biproduct Hopf $\pi $-algebra is provided. The relation between a bijective 1-cocycle and a weak Hopf $\pi $-brace is investigated.