predecessors and succesors in quivers

Let Q=(Q0,Q1,s,t) be a quiver, i.e. Q0 is a set of vertices, Q1 is a set of arrows and s,t:Q1Q0 are functions called source and target respectively. Recall, that


is a path in Q, if each αiQ1 and t(αi)=s(αi+1) for all i=1,,n-1. The length of ω is defined as n.

Definition. If a,bQ0 are vertices such that there exists a path


with s(α1)=a and t(αn)=b, then a is said to be a predecessor of b and b is said to be a successorMathworldPlanetmathPlanetmathPlanetmath of a. Additionally if there is such path of length 1, i.e. there exists an arrow from a to b, then a is a direct predecessor of b and b is a direct succesor of a.

For a given vertex aQ0 we define the following sets:

a-={bQ0|b is a direct predecessor of a};
a+={bQ0|b is a direct successor of a}.

The elements in a-a+ are called neighbours of a.

Example. Consider the following quiver:



2-={1};  2+={3,4};

and 1,3,4 are all neighbours of 2. Also 0 is a predecessor of 2, but not direct.

Title predecessors and succesors in quivers
Canonical name PredecessorsAndSuccesorsInQuivers
Date of creation 2013-03-22 19:17:47
Last modified on 2013-03-22 19:17:47
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Definition
Classification msc 14L24