  We introduce a family of order $N\in \mathbb{N}$ Lax matrices that is indexed
by the natural number $k\in \{1,\ldots,N-1\}.$ For each value of $k$ they serve
as strong Lax matrices of a hierarchy of integrable difference systems in edge
variables that in turn lead to hierarchies of integrable difference systems in
vertex variables or in a combination of edge and vertex variables. Furthermore,
the entries of the Lax matrices are considered as elements of a division ring,
so we obtain hierarchies of discrete integrable systems extended in the
non-commutative domain.
