Homomorphisms of finitely generated Abelian groups

Homomorphisms between finite Abelian groups.

class dual_pairs.abelian_group_homomorphism.AbelianGroupHomomorphism(*args: Any, **kwargs: Any)

A homomorphism between finitely generated Abelian groups.

cokernel()

Return the cokernel of self.

OUTPUT:

The cokernel of self as an Abelian group C (given by Smith normal form generators) together with a surjective homomorphism from the codomain of self to C.

EXAMPLES:

sage: from dual_pairs.abelian_group_homomorphism import hom
sage: A = AbelianGroup([2, 2, 0])
sage: B = AbelianGroup([2, 2, 0, 2, 0, 2, 0, 2, 0, 0])
sage: im_gens = (B([1, 1, 0, 1, 0, 1, 0, 0, 0, 0]),
....:            B([0, 0, 0, 0, 0, 0, 0, 1, 0, 0]),
....:            B([0, 0, 0, 0, 0, 0, 2, 0, -2, 0]))
sage: hom(A, B, im_gens).cokernel()
Abelian group morphism:
  From: Multiplicative Abelian group isomorphic to C2 x C2 x Z x C2 x Z x C2 x Z x C2 x Z x Z
  To:   Multiplicative Abelian group isomorphic to C2 x C2 x C2 x C2 x Z x Z x Z x Z
Defn:
  f0 |--> f1*f2*f3
  f1 |--> f2
  f2 |--> f4
  f3 |--> f1
  f4 |--> f5
  f5 |--> f3
  f6 |--> f6
  f7 |--> 1
  f8 |--> f0*f6
  f9 |--> f7

inverse_image(x)

Return an inverse image of x under self.

EXAMPLES:

sage: from dual_pairs.abelian_group_homomorphism import hom
sage: A = AbelianGroup([2, 4, 0])
sage: B = AbelianGroup([4, 8, 0])
sage: im_gens = (B([2, 4, 0]),
....:            B([1, 2, 0]),
....:            B([0, 4, 2]))
sage: h = hom(A, B, im_gens)
sage: y = h(A.random_element())
sage: h(h.inverse_image(y)) == y
True

kernel()

Return the kernel of self.

OUTPUT:

The kernel of self as an Abelian group K (given by Smith normal form generators) together with an injective homomorphism from K to the domain of self.

EXAMPLES:

sage: from dual_pairs.abelian_group_homomorphism import hom
sage: A = AbelianGroup([2, 2, 0])
sage: B = AbelianGroup([2, 2, 0, 2, 0, 2, 0, 2, 0, 0])
sage: im_gens = (B([1, 1, 0, 1, 0, 1, 0, 0, 0, 0]),
....:            B([0, 0, 0, 0, 0, 0, 0, 1, 0, 0]),
....:            B([0, 0, 0, 0, 0, 0, 2, 0, -2, 0]))
sage: hom(A, B, im_gens).kernel()
Abelian group morphism:
  From: Trivial Abelian group
  To:   Multiplicative Abelian group isomorphic to C2 x C2 x Z
Defn:

solve_left(h)

Return a solution g of g * self == h.

TODO: this is wrong; generalise inverse_image() instead

solve_right(h)

Return a solution g of self * g == h.

TODO: this may not always work (maybe if self is injective?)

dual_pairs.abelian_group_homomorphism.homology(f, g)

Return the homology group of the pair (f, g).

dual_pairs.abelian_group_homomorphism.solve_mod(M, N, D)

Return an integral solution of X * M == N (mod D).

EXAMPLES:

sage: from dual_pairs.abelian_group_homomorphism import solve_mod
sage: M = Matrix([(-1, -1, 0, -1, 0, 1, 0, 1, 0, 0),
....:             (-1, -1, 0, -1, 0, 1, 0, 0, 0, 0),
....:             (0, 0, 0, 0, 0, 0, 1, 0, -1, 0)])
sage: N = Matrix([(1, 1, 0, 1, 0, 1, 0, 0, 0, 0),
....:             (0, 0, 0, 0, 0, 0, 0, 1, 0, 0),
....:             (0, 0, 0, 0, 0, 0, 2, 0, -2, 0)])
sage: D = vector([2, 2, 0, 2, 0, 2, 0, 2, 0, 0])
sage: solve_mod(M, N, D)
[0 1 0]
[1 1 0]
[0 0 2]

dual_pairs.abelian_group_homomorphism.solve_mod_right(M, N, D)

Return an integral solution of M * X == N (mod D).

EXAMPLES:

sage: from dual_pairs.abelian_group_homomorphism import solve_mod_right
sage: M = Matrix([(-1, -1, 0, -1),
....:             (0, 0, 0, 0)])
sage: N = Matrix([(1, 1, 0, 1),
....:             (0, 0, 0, 0)])
sage: D = vector([2, 0])
sage: solve_mod_right(M, N, D)
[ 0  0  0  0]
[-1 -1  0 -1]
[ 0  0  0  0]
[ 0  0  0  0]