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单词 PredecessorsAndSuccesorsInQuivers
释义

predecessors and succesors in quivers


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 functions called source and target respectively. Recall, that

ω=(α1,,αn)

is a path in Q, if each αiQ1 and t(αi)=s(αi+1) for all i=1,,n-1. The length of ω is defined as n.

Definition. If a,bQ0 are vertices such that there exists a path

ω=(α1,,αn)

with s(α1)=a and t(αn)=b, then a is said to be a predecessor of b and b is said to be a successorMathworldPlanetmathPlanetmathPlanetmath 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 aQ0 we define the following sets:

a-={bQ0|b is a direct predecessor of a};
a+={bQ0|b is a direct successor of a}.

The elements in a-a+ are called neighbours of a.

Example. Consider the following quiver:

\\xymatrix&&&30\\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.

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更新时间:2025/5/4 15:22:23