  Motivated from the study of multiple ergodic average, the investigation of
multiplicative shift spaces has drawn much of interest among researchers. This
paper focuses on the relation of topologically mixing properties between
multiplicative shift spaces and traditional shift spaces. Suppose that
$\mathsf{X}_{\Omega}^{(l)}$ is the multiplicative subshift derived from the
shift space $\Omega$ with given $l > 1$. We show that
$\mathsf{X}_{\Omega}^{(l)}$ is (topologically) transitive/mixing if and only if
$\Omega$ is extensible/mixing. After introducing $l$-directional mixing
property, we derive the equivalence between $l$-directional mixing property of
$\mathsf{X}_{\Omega}^{(l)}$ and weakly mixing property of $\Omega$.
