Cleft comodules over Hopf quasigroups

In this paper, we provide necessary and sufficient conditions for a cleft right H-comodule algebra (A, rho(A)) over a Hopf quasigroup H to be isomorphic as an algebra to the crossed product A(H)sigma(AH) H, where A(H) is the coinvariants subalgebra of A and sigma(AH) is a morphism between H circle times H and A(H). As a consequence, we obtain the corresponding version in the nonassociative setting of the result given by Blattner, Cohen and Montgomery for projections of Hopf algebras with coalgebra splitting. Concrete examples satisfying the obtained conditions are provided.