  We introduce natural binary set-theoretical products on the set of all
$m$-Dyck paths, which led us to define a non-symmetric algebraic operad
$\Dy^m$, described on the vector space spanned by $m$-Dyck paths. Our
construction is closely related to the $m$-Tamari lattice, so the products
defining $\Dy^m$ are given by intervals in this lattice. For $m=1$, we recover
the notion of dendriform algebra introduced by J.-L. Loday in \cite{Lod}, and
there exists a natural operad morphism from the operad ${\mbox {\it Ass}}$ of
associative algebras into the operad $\Dy^m$, consequently $\Dy ^m$ is a Hopf
operad. We give a description of the coproduct in terms of $m$-Dyck paths in
the last section. As an additional result, for any composition of $m+1\geq 2$
with $r+1$ parts, we get a functor from the category of $\Dy ^m$ algebras into
the category of $\Dy ^r$ algebras.
