  Topological vortices in relativistic gauge theories in flat three-dimensional
spacetime are investigated. We consider the symmetry $\rm{U(1)}\times...\times
\rm{U(1)}$, and for each $\rm{U(1)}$ subgroup, a complex scalar field
transforming under its action is introduced, as well as generalized
permeabilities through which the subsystems are coupled. We investigate in
detail the features of static, finite energy solutions within this class of
generalized Maxwell-Higgs models, and study the effect of the winding numbers
in the magnetic properties of each subsystem. A BPS bound and the related first
order equations are introduced for a large class of models. Finally, we present
some specific models and solve their equations of motion to find solutions
engendering many distinct features in relation to each other and to the
standard Nielsen-Olesen vortex.
