19.65 (2018)

It is well known that varieties of groups form a free semigroup $N$ (A. L. Shmel’kin, DAN SSSR, 149 (1963), 543–545 (Russian)); B. H. Neuman, H. Neumann, P. M. Neumann, Math. Z., 80 (1962), 44–62). Varieties of linear representations over an infinite field of zero characteristic also form a free semigroup $M$, and $N$ acts freely on $M$ (B. I. Plotkin, Siberian Math. J., 13, no. 5 (1972), 713–729)

A similar theorem holds for Lie algebras (V. A. Parfenov, Algebra Logika, 6, no. 4 (1967), 61–73 (Russian); L. A. Simonyan, Siberian Math. J., 29, no. 2 (1988), 276–283). Are there other varieties of algebras $\Theta$ where the same situation takes place?