Abstract
Let HH be an algebra with a distinguished element εH∈H∗εH∈H∗ and C,DC,D two coalgebras. Based on the construction of Brzeziński’s crossed coproduct, under some suitable conditions, we introduce a coassociative coalgebra C×GTHβR×DC×TGHRβ×D which is a more general two-sided coproduct structure including two-sided smash coproduct. Necessary and sufficient conditions for C×GTHβR×DC×TGHRβ×D equipped with two-sided tensor product algebra C⊗H⊗DC⊗H⊗D to be a bialgebra (Hopf algebra) are provided. On the other hand, we obtain an improved version of the double crossed biproduct C⋆αHβ⋆DC⋆αHβ⋆D in [An extended form of Majid's double biproduct, J. Algebra Appl. 16 (4), 1760061, 2017] which induces a description of C⋆αHβ⋆DC⋆αHβ⋆D similar to Majid double biproduct C⋆H⋆DC⋆H⋆D and also present some related structures.
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