  In this paper, we study the semi-stable subcategories of the category of
representations of a Euclidean quiver, and the possible intersections of these
subcategories. Contrary to the Dynkin case, we find out that the intersection
of semi-stable subcategories may not be semi-stable. However, only a finite
number of exceptions occur, and we give a description of these subcategories.
Moreover, one can attach a simplicial fan in $\mathbb{Q}^n$ to any acyclic
quiver $Q$, and this simplicial fan allows one to completely determine the
canonical presentation of any element in $\mathbb{Z}^n$. This fan has a nice
description in the Dynkin and Euclidean cases: it is described using an
arrangement of convex codimension-one subsets of $\mathbb{Q}^n$, each such
subset being indexed by a real Schur root or a set of quasi-simple objects.
This fan also characterizes when two different stability conditions give rise
to the same semi-stable subcategory.
