# 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:

$$\text{xymatrix}\mathrm{\&}\mathrm{\&}\mathrm{\&}30\text{ar}[r]\mathrm{\&}1\text{ar}[r]\mathrm{\&}2\text{ar}[ru]\text{ar}[rd]\mathrm{\&}\mathrm{\&}\mathrm{\&}4$$ |

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 |