The object of this paper is to generalize Molnar s smash coproduct bialgebra to crossed coproduct bialgebra.

We show that C is a crossed coproduct if and only if C_R is free.

This paper gives a new method to prove the following three statements are equivalent: C/E is an H cleft coextension; C is isomorphic to a Hopf crossed coproduct E× α H with α convolution invertible; C/E is an H Galois coextension with a conormal basis property.

By using method of twisted module coalgebras, this paper shows that there are correspondings between the crossed coproducts C× α H and twisted tensor coalgebras ( CH) τ .

Moreover we give some characterizations for crossed biproducts.

This note begins with necessary and sufficient conditions for the smash product algebra A#H to be a bialgebra with a quotient bialgebra H and a weak injection from H to A#H , and points out this structure includes biproduct, bicrossed product and bicrossed coproduct as special cases by restricting maps α, β, γ or Δ A to some special maps.

This paper discussed the foundation of bicrossed products with inner action and inner coaction, investigated the conditions of a bicrossed product being to a bismash product, and proved that bismash product is isomophic to Hopf algebra HA.

The purpose of this paper is to begin to lay the foundation of bicrossed products of Hopf algebra.