Double crossed biproducts and related structures

Let $H$ be a bialgebra. $\sigma: H\otimes H\to A$ linear map, where $A$ is left $H$-comodule coalgebra, and an algebra with $H$-weak action $\triangleright$. $\tau: B$ $B$ right $\triangleleft$. In this paper, we improve the necessary conditions for two-sided crossed product $A\#^{\sigma} H~{^{\tau}\#} smash coproduct coalgebra $A\times H\times to form bialgebra (called double biproduct) such that condition $b_{[1]}\triangleright a_0\otimes b_{[0]}\triangleleft a_{-1}=a\otimes b$ in Majid's biproduct (or double-bosonization) one of conditions. On other hand, provide more general structure via Brzez\'nski's give some applications.