Basic superranks for varieties of algebras

We introduce the notion of basic superrank for varieties of algebras generalizing the notion of basic rank. First we consider a number of varieties of nearly associative algebras over a field of characteristic 0 that have infinite basic ranks and calculate their basic superranks which turns out to be finite. Namely we prove that the variety of alternative metabelian (solvable of index 2) algebras possesses the two basic superranks (1,1) and (0,3); the varieties of Jordan and Malcev metabelian algebras have the unique basic superranks (0,2) and (1,1), respectively. Furthermore, for arbitrary pair (r,s)≠(0,0) of nonnegative integers we provide a variety that has the unique basic superrank (r,s). Finally, we construct some examples of nearly associative varieties that do not possess finite basic superranks.