Meeting Covered Elements in $\nu$-Tamari Lattices

For each complete meet-semilattice $M$, we define an operator ${\mathsf{P}\mathsf{o}\mathsf{p}}_M:M\to M$ by $${\mathsf{P}\mathsf{o}\mathsf{p}}_M(x)=\bigwedge (\{y\in M:y⋖x\}\cup \{x\})\mathrm{.}$$ When $M$ is the right weak order on a symmetric group, ${\mathsf{P}\mathsf{o}\mathsf{p}}_M$ is the pop-stack-sorting map. We prove some general properties of these operators, including a theorem that describes how they interact with certain lattice congruences. We then specialize our attention to the dynamics of ${\mathsf{P}\mathsf{o}\mathsf{p}}_{\text{Tam}(\nu )}$, where $\text{Tam}(\nu )$ is the $\nu$-Tamari lattice. We determine the maximum size of a forward orbit of ${\mathsf{P}\mathsf{o}\mathsf{p}}_{\text{Tam}(\nu )}$. When $\text{Tam}(\nu )$ is the $n^{\text{th}}$ $m$-Tamari lattice, this maximum forward orbit size is $m+n-1$; in this case, we prove that the number of forward orbits of size $m+n-1$ is $$\frac{1}{n-1}\left(\genfrac{}{}{0px}{}{(m+1)(n-2)+m-1}{n-2}\right)\mathrm{.}$$ Motivated by the recent investigation of the pop-stack-sorting map, we define a lattice path $\mu \in \text{Tam}(\nu )$ to be $t$-$\mathsf{P}\mathsf{o}\mathsf{p}$-sortable if ${\mathsf{P}\mathsf{o}\mathsf{p}}_{\text{Tam}(\nu )}^t(\mu )=\nu$. We enumerate $1$-$\mathsf{P}\mathsf{o}\mathsf{p}$-sortable lattice paths in $\text{Tam}(\nu )$ for arbitrary $\nu$. We also give a recursive method to generate $2$-$\mathsf{P}\mathsf{o}\mathsf{p}$-sortable lattice paths in $\text{Tam}(\nu )$ for arbitrary $\nu$; this allows us to enumerate $2$-$\mathsf{P}\mathsf{o}\mathsf{p}$-sortable lattice paths in a large variety of $\nu$-Tamari lattices that includes the $m$-Tamari lattices.