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isClosedUnderArrows -- whether a subquiver closed under arrows

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};
i4 : SQ = Q_{0,1};
i5 : isClosedUnderArrows (SQ, Q)

o5 = true
i6 : Q = threeVertexQuiver {1, 2, 3};
i7 : SQ = Q^{0,1};
i8 : isClosedUnderArrows (SQ, Q)

o8 = true

See also

Ways to use isClosedUnderArrows:

  • isClosedUnderArrows(List,Matrix)
  • isClosedUnderArrows(List,ToricQuiver)
  • isClosedUnderArrows(Matrix,List)
  • isClosedUnderArrows(Matrix,ToricQuiver)
  • isClosedUnderArrows(ToricQuiver,ToricQuiver)

For the programmer

The object isClosedUnderArrows is a method function.


The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/ThinSincereQuivers.m2:3279:0.