  We show that suppression of vortex strings splits the order-disorder
transition in the three-state Potts ferromagnet on a simple cubic lattice and
opens up an intermediate phase characterized by partial breaking of the
three-fold symmetry and long-range order. In contrast, suppression of vortices
in the same model on a square lattice results in an intermediate phase with
enhanced U(1) symmetry and quasi-long-range order. We show that the difference
between the two phases originates from distinct patterns of domain wall
proliferation. A domain wall, separating the two most populous spin states,
percolates on its own in the former phase but remains at a percolation
threshold in the latter.
