Many of the zillions of combinatorial objects counted by Catalan numbers come with various lattice structures.
The $n$th Tamari lattice $T_n$, as originally defined, is the lattice of all those maps $f:\{1,...,n\}\to\{1,...,n\}$ which satisfy the conditions
$i\leqslant f(i)$;
if $i\leqslant j\leqslant f(i)$ then $f(j)\leqslant f(i)$,
for all $i,j\in\{1,...,n\}$, ordered valuewise, i. e. by declaring $f\leqslant g$ iff $f(i)\leqslant g(i)$ for all $i\in\{1,...,n\}$. The Hasse diagram of $T_n$ is the edge graph of the $n+1$st associahedron, tilted in a certain tricky way. Here is $T_4$ for example:
(I have omitted the last numbers of node labels, they are all $4$.)
The lattice $D_n$ of all inflationary order preserving maps $f:\{1,...,n\}\to\{1,...,n\}$, that is, maps satisfying
$i\leqslant f(i)$;
if $i\leqslant j\leqslant f(i)$ then $f(i)\leqslant f(j)$,
again ordered valuewise, has the same number of elements as $T_n$ but looks entirely different. For example, it is always distributive, while $T_n$ is not even modular for $n>2$. Here is the Hasse diagram of $D_4$:
(Last numbers are again omitted, by the same reason.)
Question: is there any natural (e. g. functorial) mapping from finite lattices to finite lattices which carries $T_n$ to $D_n$, or vice versa?
Some remarks which might or might not be relevant.
Intersection of $T_n$ and $D_n$ is the lattice of all closure operators on $\{1,...,n\}$, i. e. order preserving maps $f$ with $i\leqslant f(i)=f(f(i))$. It has $2^{n-1}$ elements and is isomorphic to the powerset of $\{1,...,n-1\}$: to a closure operator $f$ corresponds the subset of strictly $f$-increasing elements, i. e. $\{i\mid f(i)>i\}$, while to a subset $S$ corresponds the operator $f_S$ given by $f_S(i)=\min\{j\mid i\leqslant j\notin S\}$.
On the other hand, both $T_n$ and $D_n$ are embedded in the lattice $I_n$ of all inflationary operators, that is, maps $f$ with $i\leqslant f(i)$ for all $i$ - $D_n$ as a sublattice but $T_n$ only as a meet-subsemilattice. $I_n$ is distributive, it has $n!$ elements and is isomorphic to the lattice product $$ (1<2<3<\dots<n)\times(2<3<\dots<n)\times\cdots\times(n-1<n) $$ of linear orders. It is thus also natural to ask whether there is an operation that transforms the lattice $P_n$ of permutations of $\{1,...,n\}$ under the weak order into $I_n$. Like $T_n$, $P_n$ is not modular for $n>2$. Note that the Hasse diagram of $P_n$ is the edge graph of a permutohedron.
