Let be a locally finite quiver without loops. Recall that a loop is an arrow such that . Let .
Definition 1. A pair is said to be a translation quiver iff the following holds:
is a bijection;
If and is a direct predecessor of , then the number of arrows from to is equal to the number of arrows from to .
If is a translation quiver then we will say that exists if and does not exist (or it is not defined) if .
Definition 2. If is a translation quiver, then a pair is called a translation subquiver if it is a translation quiver, is a full subquiver (http://planetmath.org/SubquiverAndImageOfAQuiver) of and whenever is a vertex in such that exists and belongs to .
Example. Let be the following quiver:
If we put , and
then the pair is a translation quiver and
is its translation subquiver, where .
Remark. It is common to write translation quivers as in example. This means that is ,,oriented” to the right and in rows we have vertices such that ,,jumping” two places to the left gives us of this vertex. Note that in the example the vertex is not written in the same row as because is not (indeed, is not defined).
|Date of creation||2013-03-22 19:17:53|
|Last modified on||2013-03-22 19:17:53|
|Last modified by||joking (16130)|