  In analogy with the definition of ``extended Sobolev scale" on $\mathbb{R}^n$
by Mikhailets and Murach, working in the setting of the lattice $\mathbb{Z}^n$,
we define the ``extended Sobolev scale" $H^{\varphi}(\mathbb{Z}^n)$, where
$\varphi$ is a function which is $RO$-varying at infinity. Using the scale
$H^{\varphi}(\mathbb{Z}^n)$, we describe all Hilbert function-spaces that serve
as interpolation spaces with respect to a pair of discrete Sobolev spaces
$[H^{(s_0)}(\mathbb{Z}^n), H^{(s_1)}(\mathbb{Z}^n)]$, with $s_0<s_1$. We use
this interpolation result to obtain the mapping property and the Fredholmness
property of (discrete) pseudo-differential operators (PDOs) in the context of
the scale $H^{\varphi}(\mathbb{Z}^n)$. Furthermore, starting from a first-order
positive-definite (discrete) PDO $A$ of elliptic type, we define the ``extended
discrete $A$-scale" $H^{\varphi}_{A}(\mathbb{Z}^n)$ and show that it coincides,
up to norm equivalence, with the scale $H^{\varphi}(\mathbb{Z}^n)$.
Additionally, we establish the $\mathbb{Z}^n$-analogues of several other
properties of the scale $H^{\varphi}(\mathbb{R}^n)$.
