Varieties of bicommutative algebras with identity of degree three

TL;DR

This paper classifies varieties of bicommutative algebras satisfying degree three polynomial identities and characterizes when their subvariety lattice is distributive.

Contribution

It provides a complete description of such varieties over characteristic zero fields and establishes conditions for distributive lattice structures.

Findings

Classified all bicommutative algebra varieties with degree three identities.

Identified necessary and sufficient conditions for distributive subvariety lattices.

Enhanced understanding of the structure of bicommutative algebra varieties.

Abstract

The variety of bicommutative algebras is the class of all nonassociative algebras satisfying the polynomial identities $(x_1x_2)x_3=(x_1x_3)x_2$ and $x_1(x_2x_3)=x_2(x_1x_3)$. In this paper we provide a complete description of varieties of bicommutative algebras over a field of characteristic zero that satisfy a polynomial identity of degree three. Furthermore, we establish a sufficient and necessary condition for a variety of bicommutative algebras to have a distributive lattice of subvarieties.