Substructures of bi-Frobenius algebras

Bi-Frobenius algebras (or bF algebras) were recently introduced by the author and Takeuchi. These are both Frobenius algebras and Frobenius coalgebras and satisfy some compatibility conditions. The concept generalizes finite dimensional Hopf algebras. In Section 1 we give conditions for finite dimensional algebras and coalgebras to be bF algebras. In Section 2 we discuss substructures, quotient structures of bF algebras. Section 3 is devoted a study of morphisms and we deduce some results in Koppinenʹs theory.