Discrete Lax pairs and hierarchies of integrable difference systems

Abstract: 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.