  In this article we establish theory of semi-orthogonal Parseval wavelets
associated to generalized multiresolution analysis (GMRA) for the local field
of positive characteristics (LFPC). By employing the properties of translation
invariant spaces on the core space of GMRA we obtain a characterization of
semi-orthogonal Parseval wavelets in terms of consistency equation for LFPC. As
a consequence, we obtain a characterization of an orthonormal (multi)wavelet to
be associated with an MRA in terms of multiplicity function as well as
dimension function of a (multi)wavelet. Further, we provide characterizations
of Parseval scaling functions, scaling sets and bandlimited wavelets together
with a Shannon type multiwavelet for LFPC.
