  Motivated by the fundamental results of the geometric algebra we study
quadrilateral lattices in projective spaces over division rings. After giving
the noncommutative discrete Darboux equations we discuss differences and
similarities with the commutative case. Then we consider the fundamental
transformation of such lattices in the vectorial setting and we show the
corresponding permutability theorems. We discuss also the possibility of
obtaining in a similar spirit a noncommutative version of the B-(Moutard)
quadrilateral lattices.
