  For any finite path $v$ on the square grid consisting of north and east unit
steps, starting at (0,0), we construct a poset Tam$(v)$ that consists of all
the paths weakly above $v$ with the same number of north and east steps as $v$.
For particular choices of $v$, we recover the traditional Tamari lattice and
the $m$-Tamari lattice.
  Let $\overleftarrow{v}$ be the path obtained from $v$ by reading the unit
steps of $v$ in reverse order, replacing the east steps by north steps and vice
versa. We show that the poset Tam$(v)$ is isomorphic to the dual of the poset
Tam$(\overleftarrow{v})$. We do so by showing bijectively that the poset
Tam$(v)$ is isomorphic to the poset based on rotation of full binary trees with
the fixed canopy $v$, from which the duality follows easily. This also shows
that Tam$(v)$ is a lattice for any path $v$. We also obtain as a corollary of
this bijection that the usual Tamari lattice, based on Dyck paths of height
$n$, is a partition of the (smaller) lattices Tam$(v)$, where the $v$ are all
the paths on the square grid that consist of $n-1$ unit steps.
  We explain possible connections between the poset Tam$(v)$ and (the
combinatorics of) the generalized diagonal coinvariant spaces of the symmetric
group.
