In mathematics, a bialgebra over a field K is a structure which is both a unital associative algebra and a coalgebra over K, such that the comultiplication and the counit are both unital algebra homomorphisms. Equivalently, one may require that the multiplication and the unit of the algebra both be coalgebra morphisms. The compatibility conditions can also be expressed by the following commutative diagrams:

Here ∇ : B ⊗ B → B is the algebra multiplication and η : K → B is the unit of the algebra. Δ : B → B ⊗ B is the comultiplication and ε : B → K is the counit. τ : B ⊗ B → B ⊗ B is the linear map defined by τ(x⊗y) = y⊗x for all x and y in B.

In formulas, the bialgebra compatibility conditions look as follows (using the sumless Sweedler notation):

Here we wrote the algebra multiplication as simple juxtaposition, and 1 is the multiplicative identity.

For examples of bialgebras, refer to the articles on coalgebras and Hopf algebras. (Hopf algebras are bialgebras with certain additional structure.)