On Hopf algebras with positive bases

We show that if a finite dimensional Hopf algebra H over C has a basis with respect to which all the structure constants are nonnegative, then H is isomorphic to the bi-cross-product Hopf algebra constructed by Takeuchi and Majid from a finite group G and a unique factorization G=G₊G_- of G into two subgroups. We also show that Hopf algebras in the category of finite sets with correspondences as morphisms are classified in a similar way. Our results can be used to explain some results on Hopf algebras from the set-theoretical point of view.