# isClosedUnderArrows -- is a subquiver closed under arrows?

## Synopsis

• Usage:
isClosedUnderArrows (V, Q)
isClosedUnderArrows (M, Q)
isClosedUnderArrows (SQ, Q)
• Inputs:
• M, , incidence matrix of subquiver to check
• Q, an instance of the type ToricQuiver,
• SQ, an instance of the type ToricQuiver, subquiver of Q to check
• V, a list, set of vertices
• Outputs:

## Description

checks that a set of vertices is closed under arrows with respect to the toricQuiver Q. That is, for any $v\in V$, then any arrow in $Q_1$ with tail $v$ must have head in $V$ as well. Note that this does not require that $V\subset Q_0$.

Note also that the attribute closed under arrows relates to the underlying graph. Arrows with flow of 0 (which occur in cases where using the quiver subset form: Q^S rather than Q_S) are considered as valid arrows.

 i1 : isClosedUnderArrows ({0, 2, 3}, bipartiteQuiver(2, 3)) o1 = false i2 : isClosedUnderArrows ({2, 3, 4}, bipartiteQuiver(2, 3)) o2 = true
 i3 : Q = threeVertexQuiver {1, 2, 3} o3 = ToricQuiver{flow => {1, 1, 1, 1, 1, 1} } IncidenceMatrix => | -1 -1 -1 -1 0 0 | | 1 0 0 0 -1 -1 | | 0 1 1 1 1 1 | Q0 => {0, 1, 2} Q1 => {{0, 1}, {0, 2}, {0, 2}, {0, 2}, {1, 2}, {1, 2}} weights => {-4, -1, 5} o3 : ToricQuiver i4 : SQ = Q_{0,1} o4 = ToricQuiver{flow => {1, 1} } IncidenceMatrix => | -1 -1 | | 1 0 | | 0 1 | Q0 => {0, 1, 2} Q1 => {{0, 1}, {0, 2}} weights => {-2, 1, 1} o4 : ToricQuiver i5 : isClosedUnderArrows (SQ, Q) o5 = true
 i6 : Q = threeVertexQuiver {1, 2, 3} o6 = ToricQuiver{flow => {1, 1, 1, 1, 1, 1} } IncidenceMatrix => | -1 -1 -1 -1 0 0 | | 1 0 0 0 -1 -1 | | 0 1 1 1 1 1 | Q0 => {0, 1, 2} Q1 => {{0, 1}, {0, 2}, {0, 2}, {0, 2}, {1, 2}, {1, 2}} weights => {-4, -1, 5} o6 : ToricQuiver i7 : SQ = Q^{0,1} o7 = ToricQuiver{flow => {1, 1, 0, 0, 0, 0} } IncidenceMatrix => | -1 -1 -1 -1 0 0 | | 1 0 0 0 -1 -1 | | 0 1 1 1 1 1 | Q0 => {0, 1, 2} Q1 => {{0, 1}, {0, 2}, {0, 2}, {0, 2}, {1, 2}, {1, 2}} weights => {-2, 1, 1} o7 : ToricQuiver i8 : isClosedUnderArrows (SQ, Q) o8 = true

## Ways to use isClosedUnderArrows :

• "isClosedUnderArrows(List,ToricQuiver)"
• "isClosedUnderArrows(Matrix,ToricQuiver)"
• "isClosedUnderArrows(ToricQuiver,ToricQuiver)"
• isClosedUnderArrows(List,Matrix) (missing documentation)
• isClosedUnderArrows(Matrix,List) (missing documentation)

## For the programmer

The object isClosedUnderArrows is .