Basic superranks for varieties of algebras

TL;DR

This paper introduces the concept of basic superrank for algebra varieties, computes it for several nearly associative varieties, and constructs examples with infinite superranks, expanding understanding of algebraic hierarchy.

Contribution

It generalizes the notion of basic rank to basic superrank, calculates it for various algebra varieties, and provides examples with both finite and infinite superranks.

Findings

Alternative metabelian algebras have superranks (1,1) and (0,3)

Jordan and Malcev metabelian algebras have superranks (0,2) and (1,1)

Existence of varieties with arbitrary superranks and some with infinite superranks

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.