  Since its introduction by Loday in 1995 with motivation from algebraic
K-theory, dendriform dialgebras have been studied quite extensively with
connections to several areas in mathematics and physics. A few more similar
structures have been found recently, such as the tri-, quadri-, ennea- and
octo-algebras, with increasing complexity in their constructions and
properties. We consider these constructions as operads and their products and
duals, in terms of generators and relations, with the goal to clarify and
simplify the process of obtaining new algebra structures from known structures
and from linear operators.
