  We construct dyonic states in 2+1-dimensional lattice Z_N-Higgs models, i.e.,
states which are both, electrically and magnetically charged. The associated
Hilbert spaces carry charged representations of the observable algebra, the
global transfer matrix and a unitary implementation of the group of spatial
lattice translations. We prove that for coinciding total charges these
representations are dynamically equivalent and we construct a local intertwiner
connection depending on a path in the space of charge distributions. The
holonomy of this connection is given by Z_N-valued phases. This will be the
starting point for a construction of scattering states with anyon statistics in
a subsequent paper.
