$(P,\phi )$-Tamari lattices

Given any poset $P$ and chain $\phi$ in $P$, we define the $(P,\phi )$-Tamari lattice. We study in depth these lattices and prove in particular that they are join-semidistributive, join-congruence uniform and left modular. We prove that the lattices of higher torsion classes of the higher Auslander and Nakayama algebras of type $\mathbb{A}$ are examples of $(P,\phi )$-Tamari lattices and thus they inherit their properties. We also give general results related to left modular, extremal and congruence normal lattices.