We study a compatibility relationship between Qin's dominance order on a cluster algebra A and partial orderings arising from classifications of simple objects in a monoidal categorification C of A. Our motivating example is Hernandez-Leclerc's monoidal categorification using representations of quantum affine algebras. In the framework of Kang-Kashiwara-Kim-Oh's monoidal categorification via representations of quiver Hecke algebras, we focus on the case of the category R´gmod for a symmetric finite type A n quiver Hecke algebra using Kleshchev-Ram's classification of irreducible finite-dimensional representations.