  We study a compatibility relationship between Qin's dominance order on a
cluster algebra $\mathcal{A}$ and partial orderings arising from
classifications of simple objects in a monoidal categorification $\mathcal{C}$
of $\mathcal{A}$. 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 $A$ quiver Hecke algebra using
Kleshchev-Ram's classification of irreducible finite dimensional
representations.
