In this paper we deal with the of an $MV$-algebra $𝒜$, where $\alpha$ and $\beta$ are nonzero cardinals. It is proved that if $𝒜$ is singular and $(\alpha ,2)$-distributive, then it is . We show that if $𝒜$ is complete then it can be represented as a direct product of $MV$-algebras which are homogeneous with respect to higher degrees of distributivity.