  We explore the Potts model on the generalized decorated square lattice, with
both nearest (J1) and next-neighbor (J2) interactions. Using the tensor
renormalization-group method augmented by higher-order singular value
decompositions, we calculate the spontaneous magnetization of the Potts model
with q = 2, 3, and 4. The results for q = 2 allow us to benchmark our numerics
using the exact solution. For q = 3, we find a highly degenerate ground state
with partial order on a single sublattice, but with vanishing entropy per site,
and we obtain the phase diagram as a function of the ratio J2=J1. There is no
finite-temperature transition for the q = 4 case when J1 = J2, but the magnetic
susceptibility diverges as the temperature goes to zero, showing that the model
is critical at T = 0.
