  The internal degrees of freedom provided by ultracold atoms give a route for
realizing higher dimensional physics in systems with limited spatial
dimensions. Non-spatial degrees of freedom in these systems are dubbed
"synthetic dimensions". This connection is useful from an experimental
standpoint but complicated by the fact that interactions alter the condensate
ground state. Here we use the Gross-Pitaevskii equation to study ground state
properties of a spin-1 Bose gas under the combined influence of an optical
lattice, spin-orbit coupling, and interactions at the mean field level. The
associated phases depend on the sign of the spin-dependent interaction
parameter and the strength of the optical lattice potential. We find "charge"
and spin density wave phases which are directly related to helical spin order
in real space and affect the behavior of edge currents in the synthetic
dimension. We determine the resulting phase diagram as a function of the
spin-orbit coupling and spin-dependent interaction strength, considering both
attractive (ferromagnetic) and repulsive (polar) spin-dependent interactions.
Our results are applicable to current and future experiments, specifically with
$^{87}$Rb, $^{7}$Li, $^{41}$K, and $^{23}$Na.
