Given a hereditary (not sure if needed) algebra $A$ (fin.dim. over a field and connected, maybe also assume representation-finite) and let $X$ be the set of $A$-modules with endomorphism ring isomorphic to $A$. For $M,N$ in $X$, define $M \geq N$ iff ($M =N$ or $dim(Hom_A(M,N))>dim(Hom_A(N,M))$) .
Is this order a partial order (that is transitive)? I have no experience with this and my Pc is too slow to test many examples. However, it seems to be true for hereditary Nakayama algebras with less than or equal 4 simples.
(maybe a better question would be: When is this order transitive?)
Is this a partial order for other interesting sets $X$, maybe tilting modules or something like that?
