ToricMap == ToricMap -- whether to toric maps are equal

Synopsis

Operator: ==
Usage:

f == g
Inputs:
- f, a toric map,
- g, a toric map,
Outputs:
- a Boolean value, that is true if the toric maps have the same source, same target, and same underlying map of lattices

Two toric maps are equal if their three defining attributes (namely source, target, and underlying matrix) are the same.

We illustrate this test with the projection from a blow-up at a point in the projective plane to the projective plane and various identity maps.

i1 : Y = toricProjectiveSpace 2;

i2 : X = toricBlowup({0, 2}, Y);

i3 : f = X^[]

o3 = | 1 0 |
     | 0 1 |

o3 : ToricMap Y <--- X

i4 : assert (isWellDefined f and f == map(Y, X, 1))

i5 : g = id_X

o5 = | 1 0 |
     | 0 1 |

o5 : ToricMap X <--- X

i6 : assert (g == map(X, X, 1))

i7 : assert (f != g)

i8 : assert (isWellDefined g and source g === X and target g === X)

i9 : assert (matrix f == matrix g and source f === source g and
         target f =!= target g)

The second example shows that we can have more than one well-defined toric map with the same source and target.

i10 : Z = toricProjectiveSpace 1;

i11 : pi1 = map(Z, X, matrix{{0, 1}})

o11 = | 0 1 |

o11 : ToricMap Z <--- X

i12 : assert (isWellDefined pi1 and source pi1 === X and target pi1 === Z)

i13 : pi2 = map(Z, X, matrix{{0, 2}})

o13 = | 0 2 |

o13 : ToricMap Z <--- X

i14 : assert (isWellDefined pi2 and source pi2 === X and target pi2 === Z)

i15 : assert (pi1 != pi2)