predecessors and succesors in quivers
Let be a quiver, i.e. is a set of vertices, is a set of arrows and are functions called source and target respectively. Recall, that
is a path in , if each and for all . The length of is defined as .
Definition. If are vertices such that there exists a path
with and , then is said to be a predecessor of and is said to be a successor of . Additionally if there is such path of length , i.e. there exists an arrow from to , then is a direct predecessor of and is a direct succesor of .
For a given vertex we define the following sets:
The elements in are called neighbours of .
Example. Consider the following quiver:
and are all neighbours of . Also is a predecessor of , but not direct.
|Title||predecessors and succesors in quivers|
|Date of creation||2013-03-22 19:17:47|
|Last modified on||2013-03-22 19:17:47|
|Last modified by||joking (16130)|