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The smash product algebra has been generalized to general smash product algebra in  and we can generalize the smash coproduct coalgebra to obtain the general smash coproduct coalgebra. It is natural to replace the smash product and smash coproduct by the generalized smash product and generalized smash coproduct and consider the condition under which the generalized smash product algebra structure and the generalized smash coproduct coalgebra structure will inherit a bialgebra structure or a Hopf algebra structure. We derive necessary suﬃcient conditions for the problem. This generalizes the corresponding results in  and .
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