predecessors and succesors in quivers

Let $Q=(Q_0,Q_1,s,t)$ be a quiver, i.e. $Q_0$ is a set of vertices, $Q_1$ is a set of arrows and $s,t:Q_1\to Q_0$ are functions called source and target respectively. Recall, that

$$\omega =({\alpha }_1,\mathrm{\dots },{\alpha }_n)$$

is a path in $Q$, if each ${\alpha }_i\in Q_1$ and $t({\alpha }_i)=s({\alpha }_{i+1})$ for all $i=1,\mathrm{\dots },n-1$. The length of $\omega$ is defined as $n$.

Definition. If $a,b\in Q_0$ are vertices such that there exists a path

$$\omega =({\alpha }_1,\mathrm{\dots },{\alpha }_n)$$

with $s({\alpha }_1)=a$ and $t({\alpha }_n)=b$, then $a$ is said to be a predecessor of $b$ and $b$ is said to be a successor 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 $a\in Q_0$ we define the following sets:

$$a^{-}=\{b\in Q_0|b\text{ is a direct predecessor of }a\};$$

$$a^{+}=\{b\in Q_0|b\text{ is a direct successor of }a\}.$$

The elements in $a^{-}\cup a^{+}$ are called neighbours of $a$.

Example. Consider the following quiver:

Then

$$2^{-}=\{1\};{\mathrm{\hspace{0.25em}\hspace{0.25em}2}}^{+}=\{3,4\};$$

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