Mobius inversion on a multiplicative semigroup and inverse lattice problems in physics

The generalized Mobius function and Mobius inversion formula are applied to a multiplicative semigroup. A general mathematical method based on this Mobius inversion is presented to solve inversion problems of expansions with unequally weighted terms. By this method. all the inverse lattice problems in physics can be solved concisely. The solutions of four inverse lattice problems: the Fibonacci structure, the square lattice structure, the bce and the hcp lattice structures are given. These are difficult to be solved by other methods. (C) 1998 by B. G. Teubner Stuttgart-John Wiley & Sons Ltd.