The poset of positive roots and its relatives
  Let $\Delta$ be a root system with a subset of positive roots, $\Delta^+$. We consider edges of the Hasse diagrams of some posets associated with $\Delta^+$. For each edge one naturally defines its type, and we study the partition of the set of edges into types. For $\Delta^+$, the type is a simple root, and for the posets of ad-nilpotent and Abelian ideals the type is an affine simple roots. We give several descriptions of the set of edges of given type and uniform expressions for the number of edges. By a result of Peterson, the number of Abelian ideals is $2^n$, where $n$ is the rank of $\Delta$. We prove that the number of edges of the corresponding Hasse diagram is $(n+1)2^{n-2}$. For $\Delta^+$ and the Abelian ideals, we compute the number of edges of each type and prove that the number of edges of type $\alpha$ depends only on the length of $\alpha$. 
Dmitri I. Panyushev