Basic superranks for varieties of algebras

Abstract: We introduce the notion of basic superrank for varieties of algebras which generalizes that 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 has 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 $(r,s)\neq (0,0)$ 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.