Radford $[(m,k),m]$-biproduct Theorem for Generalized Hom-crossed Products

In this paper, we mainly provide a new approache to construct Hom-Hopf algebras. For this, we introduce and study the notion of a left $(m,k)$-Hom-crossed product structure as a generalization of $k$-Hom-smash product structure. Then one combines this $(m,k)$-Hom-crossed product structure and a left $m$-Hom-smash coproduct structure to build Radford $[(m,k),m]$-biproduct theorem. Finally, we study Hom admissible mappping system to characterize this Radford $[(m,k),m]$-biproduct structure.