Crossed product and Galois extension of monoidal Hom-Hopf algebras

Let $(H,\alpha )$ be a monoidal Hom-Hopf algebra, and $(A,\beta )$ a Hom-algebra. In this paper we will introduce the crossed product $(A{\mathrm{\#}}_{\sigma }H,\beta \otimes \alpha )$, which is a Hom-algebra. Then we will introduce the notions of cleft extensions and Galois extensions respectively, and prove that a crossed product is equivalent to a cleft extension and a cleft extension is equivalent to a Galois extension with normal bases property.