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 |
The object isClosedUnderArrows is a method function.