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isClosedUnderArrows -- whether a subquiver closed under arrows

Synopsis

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}}
                 synonym => toric quiver
                 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}}
                 synonym => toric quiver
                 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}}
                 synonym => toric quiver
                 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}}
                 synonym => toric quiver
                 weights => {-2, 1, 1}

o7 : ToricQuiver
 
i8 : isClosedUnderArrows (SQ, Q)

o8 = true

See also

Ways to use isClosedUnderArrows :

For the programmer

The object isClosedUnderArrows is a method function.