4-type subvarieties of the variety of associative algebras

In this paper, we consider four types of subvarieties of the variety of associative algebras. We study these subvarieties from the point of view of operads and show their connections with well-known classes of algebras, such as dendriform algebras and noncommutative Novikov algebras. Finally, we define the commutator and anti-commutator operations on these algebras and derive several identities satisfied by these operations.