A general construction of family algebraic structures

We give a general account of family algebras over a finitely presented linear operad. In a family algebra, each operation of arity n is replaced by a family of operations indexed by \(\Omega ^n\), where \(\Omega \) is a set of parameters. We show that the operad, together with its presentation, naturally defines an algebraic structure on the set of parameters, which in turn is used in the description of the family version of the relations between operations. The examples of dendriform and duplicial family algebras (hence with two parameters) and operads are treated in detail, as well as the pre-Lie family case. Finally, free one-parameter duplicial family algebras are described, together with the extended duplicial semigroup structure on the set of parameters.