  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. Performing their reduction to the
one-dimensional case by imposing suitable constraints we derive the
corresponding 1D fermionic TL hierarchies. We develop the generalized graded
R-matrix formalism using the generalized graded bracket on the space of graded
operators with an 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. We propose the graded modified Yang-Baxter
equation in the operator form and demonstrate that for the class of graded
antisymmetric R-matrices it is equivalent to the tensor form of the graded
classical Yang-Baxter equation.
