  A theory of higher rank multiresolution analysis is given in the setting of
abelian multiscalings. This theory enables the construction, from a higher rank
MRA, of finite wavelet sets whose multidilations have translates forming an
orthonormal basis in $L^2(\bR^d)$. While tensor products of uniscaled MRAs
provide simple examples we construct many nonseparable higher rank wavelets. In
particular we construct Latin square wavelets as rank 2 variants of Haar
wavelets. Also we construct nonseparable scaling functions for rank 2 variants
of Meyer wavelet scaling functions, and we construct the associated
nonseparable wavelets with compactly supported Fourier transforms. On the other
hand we show that compactly supported scaling functions for biscaled MRAs are
necessarily separable.
