locally finite quiver

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 source and target functions.

Definition. We will say that Q is locally finitePlanetmathPlanetmath iff for any vertex aQ0 there is a finite number of neighbours (http://planetmath.org/PredecessorsAndSuccesorsInQuivers) of a. Equivalently if there is a finite number of arrows ending in a and finite number of arrows starting from a.

Note that even when Q has a finite number of vertices, then Q is not necessarily locally finite. Consider the following example:


such that s(n)=t(n)=* for any n. In other words Q has one vertex and countably many arrows starting and ending at it. This quiver is not locally finite.

Every finite quiver, i.e. quiver with finite number of vertices and arrows is locally finite.

Title locally finite quiver
Canonical name LocallyFiniteQuiver
Date of creation 2013-03-22 19:17:51
Last modified on 2013-03-22 19:17:51
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Definition
Classification msc 14L24