  In this paper, we introduce and study a class of pseudo-differential
operators on the lattice $\mathbb{Z}^n$. More preciously, we consider a
weighted symbol class $M_{\rho, \Lambda}^m(\mathbb{ Z}^n\times \mathbb{T}^n),
m\in \mathbb{R}$ associated to a suitable weight function $\Lambda$ on
$\mathbb{ Z }^n$. We study elements of the symbolic calculus for
pseudo-differential operators associated with $M_{\rho, \Lambda}^m(\mathbb{
Z}^n\times \mathbb{T}^n)$ by deriving formulae for the composition, adjoint,
transpose. We define the notion of $M$-ellipticity for symbols belonging to
$M_{\rho, \Lambda}^m(\mathbb{ Z}^n\times \mathbb{T}^n)$ and construct the
parametrix of $M$-elliptic pseudo-differential operators. Further, we
investigate the minimal and maximal extensions for $M$-elliptic
pseudo-differential operators and show that they coincide on
$\ell^2(\mathbb{Z}^n)$ subject to the $M$-ellipticity of symbols. We also
determine the domains of the minimal and maximal operators. Finally, We discuss
Fredholmness and compute the index of $M$-elliptic pseudo-differential
operators on $\mathbb{Z}^n$.
