  We introduce a new poset structure on Dyck paths where the covering relation
is a particular case of the relation inducing the Tamari lattice. We prove that
the transitive closure of this relation endows Dyck paths with a lattice
structure. We provide a trivariate generating function counting the number of
Dyck paths with respect to the semilength, the numbers of outgoing and incoming
edges in the Hasse diagram. We deduce the numbers of coverings, meet and join
irreducible elements. As a byproduct, we present a new involution on Dyck paths
that transports the bistatistic of the numbers of outgoing and incoming edges
into its reverse. Finally, we give a generating function for the number of
intervals, and we compare this number with the number of intervals in the
Tamari lattice.
