$(P,\ddot{I}•)$-Tamari lattices

Abstract

Given any poset $P$ and chain $\ddot{I}•$ in $P$, we define the $(P,\ddot{I}•)$-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 $(P,\ddot{I}•)$-Tamari lattices and thus they inherit their properties. We also give general results related to left modular, extremal and congruence normal lattices.

This paper is an extended abstract of 12 pages that was accepted for FPSAC 2025