# 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

 $\omega=(\alpha_{1},\ldots,\alpha_{n})$

is a path in $Q$, if each $\alpha_{i}\in Q_{1}$ and $t(\alpha_{i})=s(\alpha_{i+1})$ for all $i=1,\ldots,n-1$. The length of $\omega$ is defined as $n$.

If $a,b\in Q_{0}$ are vertices such that there exists a path

 $\omega=(\alpha_{1},\ldots,\alpha_{n})$

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 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:

 $a^{-}=\{b\in Q_{0}\ |\ b\mbox{ is a direct predecessor of }a\};$
 $a^{+}=\{b\in Q_{0}\ |\ b\mbox{ is a direct successor of }a\}.$

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

Example. Consider the following quiver:

 $\xymatrix{&&&3\\ 0\ar[r]&1\ar[r]&2\ar[ru]\ar[rd]\\ &&&4}$

Then

 $2^{-}=\{1\};\ \ 2^{+}=\{3,4\};$

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 PredecessorsAndSuccesorsInQuivers 2013-03-22 19:17:47 2013-03-22 19:17:47 joking (16130) joking (16130) 4 joking (16130) Definition msc 14L24