Compatible, split and family Loday-algebras

Abstract

Given a nonsymmetric operad O, we first construct two new nonsymmetric operads O comp and O Dend . These operads are respectively useful to study compatible and split Loday-algebras. As an application of the operad O comp , we show that the cohomology of a compatible associative algebra carries a Gerstenhaber structure. We give an application of the operad O Dend to dendriform algebras and find generalizations to other Loday-algebras. In the end, we construct another operad Fam(O Ω ) Dend to study dendriform-family algebras recently introduced in the literature. We also define and study homotopy dendriform-family algebras.