Varieties of bicommutative algebras with identity of degree three

The variety of bicommutative algebras is the class of all nonas-sociative algebras satisfying the polynomial identities (x 1 x 2 )x 3 = (x 1 x 3 )x 2 and x 1 (x 2 x 3 ) = x 2 (x 1 x 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.