quotient quiver

Let Q=(Q0,Q1,s,t) be a quiver.

Definition. An equivalence relationMathworldPlanetmath on Q is a pair


such that 0 is an equivalence relation on Q0, 1 is an equivalence relation on Q1 and if


for some arrows α,βQ1, then

s(α)0s(β) and t(α)1t(β).

If is an equivalence relation on Q, then (Q0/0,Q1/1,s,t) is a quiver, where

s([α])=[s(α)]   t([α])=[t(α)].

This quiver is called the quotient quiver of Q by and is denoted by Q/.

It can be easily seen, that if Q is a quiver and is an equivalence relation on Q, then


given by π=(π0,π1), where π0 and π1 are quotient maps is a morphismMathworldPlanetmathPlanetmath of quivers. It will be called the quotient morphism.

Example. Consider the following quiver


If we take by putting 204 and a1b, c1d, then the corresponding quotient quiver is isomorphic to

Title quotient quiver
Canonical name QuotientQuiver
Date of creation 2013-03-22 19:17:22
Last modified on 2013-03-22 19:17:22
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Definition
Classification msc 14L24