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.

We give two variations of the Holland representation theorem for $\ell$-groups and of its generalization of Glass for directed interpolation po-groups as groups of automorphisms of a linearly ordered set or of an antilattice, respectively. We show that every pseudo-effect algebra with some kind of the Riesz decomposition property as well as any pseudo $MV$-algebra can be represented as a pseudo-effect algebra or as a pseudo $MV$-algebra of automorphisms of some antilattice or of some linearly ordered set.

The variety of basic algebras is closed under formation of horizontal sums. We characterize when a given basic algebra is a horizontal sum of chains, MV-algebras or Boolean algebras.