  By exhibiting the corresponding Lax pair representations we propose a wide
class of integrable two-dimensional (2D) fermionic Toda lattice (TL)
hierarchies which includes the 2D N=(2|2) and N=(0|2) supersymmetric TL
hierarchies as particular cases. We develop the generalized graded R-matrix
formalism using the generalized graded bracket on the space of graded operators
with involution generalizing the graded commutator in superalgebras, which
allows one to describe these hierarchies in the framework of the Hamiltonian
formalism and construct their first two Hamiltonian structures. The first
Hamiltonian structure is obtained for both bosonic and fermionic Lax operators
while the second Hamiltonian structure is established for bosonic Lax operators
only.
