  Suppose $\Gamma$ is a submonoid of a lattice, not containing a line. In this
note, we use the natural $\Gamma$-grading on the monoid algebra $R[\Gamma]$ to
prove structural results about the relative $K$-theory $K(R[\Gamma], R)$. When
$R$ contains a field, we prove a decomposition indexed by the rays in $\Gamma$,
and a compatible action by the Witt vectors of $R$ for each $\mathbf N$-grading
of $\Gamma$. In characteristic zero, there is additionally an action by Witt
vectors for the truncation set $\Gamma$. Finally, we apply this to get a
ray-like description of $K_*(R[x_1,...,x_n])$ proposed by J.\,Davis.
