  In development of the started activity on lattice analogues of $W$-algebras,
we define the notion of lattice $W_{\infty}$-algebra, accociated with lattice
integrable system with infinite set of fields. Various kinds of reduction to
lattice $W_N$-algebras, related to discrete $N$-KdV hierarchies are described.
We also discuss the connection of our results with those obtained in the papers
of Xiong [13] and Bonora [14].
