Noncommutative geometry on a discrete periodic lattice and gauge theory

We discuss the quantum mechanics of a particle in a magnetic field when its position x(mu) is restricted to a periodic lattice, while its momentum p(mu) is restricted to a periodic dual lattice. Through these considerations we define non-commutative geometry on the lattice. This leads to a deformation of the algebra of functions on the lattice, such that their product involves a "diamond" product, which becomes the star product in the continuum limit. We apply these results to construct non-commutative U(1) and U(M) gauge theories, and show that they are equivalent to a purr U(NM) matrix theory, where N-2 is the number of lattice points.