ABSTRACT
We will consider quotients of the free nonsymmetric operad T (X ) by several relations of weight 3. Similarly to how operads with relations of weight 2 are conventionally referred to as quadratic, operads with relations of weight 3 are called cubic. In a more classical language of identities in nonassociative algebra, we focus on identities of arity 4 that involve one binary operation and are nonsymmetric, so that in nonassociative monomials of each identity all arguments appear in the same order (like in the associativity identity). Our methods in principle apply to any number of operations, of any arities, satisfying relations of any arities, either symmetric or nonsymmetric. However, as we shall see below, sizes of matrices involved in investigating these questions grow very fast, so computational feasibility may be an issue in practice.
