5.44 (1976)

A union $\mathfrak{A} = \bigcup_{\alpha} \mathfrak{V}_\alpha$ of varieties of groups (in the lattice of varieties) is called irreducible if $\bigcup_{\beta \neq \alpha} \mathfrak{V}_\beta \neq \mathfrak{A}$ for each index $\alpha$. Is every variety an irreducible union of (finitely or infinitely many) varieties each of which cannot be decomposed into a union of two proper subvarieties?