  We introduce a notion of split extension of (non-associative) bialgebras
which generalizes the notion of split extension of magmas introduced by M.
Gran, G. Janelidze and M. Sobral. We show that this definition is equivalent to
the notion of action of (non-associative) bialgebras. We particularize this
equivalence to (non-associative) Hopf algebras by defining split extensions of
(non-associative) Hopf algebras and proving that they are equivalent to actions
of (non-associative) Hopf algebras. Moreover, we prove the validity of the
Split Short Five Lemma for these kinds of split extensions, and we examine some
examples.
