  Differential structure of a d-dimensional lattice, which is essentially a
noncommutative exterior algebra, is defined using reductions in first order and
second order of universal differential calculus in the context of
noncommutative geometry (NCG) developed by Dimakis et al. This differential
structure can be realized adopting a Dirac-Connes operator proposed by us
recently within Connes' NCG. With matrix representations being specified, our
Dirac-Connes operator corresponds to staggered Dirac operator, in the case that
dimension of the lattice equals to 1, 2 and 4.
