Cliff operads: a hierarchy of operads on words

A new hierarchy of operads over the linear spans $$\delta $$ -cliffs, which are some words integers, is introduced. These intended to be analogues operad permutations, also known as associative symmetric operad. We obtain whose partial compositions can described in terms intervals lattice -cliffs. very peculiar world combinatorial since, despite relative simplicity for their construction, they infinitely generated and have nonquadratic nonhomogeneous nontrivial relations. provide a general construction quotients. use it endow spaces m-increasing trees, c-rectangular paths, m-Dyck paths with structures. The on admit, Koszul duals, generalizing duplicial triplicial operads.