Lattice Properties of Oriented Exchange Graphs and Torsion Classes

摘要: The exchange graph of a 2-acyclic quiver is the mutation-equivalent quivers whose edges correspond to mutations. When admits nondegenerate Jacobi-finite potential, natural acyclic orientation called oriented graph, as shown by Brustle and Yang. isomorphic Hasse diagram poset functorially finite torsion classes certain dimensional algebra. We prove that lattices are semidistributive lattices, we use this result conclude graphs with finitely many elements lattices. Furthermore, if type A Dynkin or an cycle, then lattice quotient biclosed subcategories modules over cluster-tilted algebra, generalizing Reading's Cambrian in A. also apply our results address conjecture Brustle, Dupont, Perotin on lengths maximal green sequences.