  Generalizing the graded commutator in superalgebras, we propose a new bracket
operation on the space of graded operators with an involution. We study
properties of this operation and show that the Lax representation of the
two-dimensional N=(1|1) supersymmetric Toda lattice hierarchy can be realized
via the generalized bracket operation; this is important in constructing the
semiclassical (continuum) limit of this hierarchy. We construct the continuum
limit of the N=(1|1) Toda lattice hierarchy, the dispersionless N=(1|1) Toda
hierarchy. In this limit, we obtain the Lax representation, with the
generalized graded bracket becoming the corresponding Poisson bracket on the
graded phase superspace. We find bosonic symmetries of the dispersionless
N=(1|1) supersymmetric Toda equation.
