# Multiplicative Abelian Groups With Values#

Often, one ends up with a set that forms an Abelian group. It would be nice if one could return an Abelian group class to encapsulate the data. However, AbelianGroup() is an abstract Abelian group defined by generators and relations. This module implements AbelianGroupWithValues that allows the group elements to be decorated with values.

An example where this module is used is the unit group of a number field, see sage.rings.number_field.unit_group. The units form a finitely generated Abelian group. We can think of the elements either as abstract Abelian group elements or as particular numbers in the number field. The AbelianGroupWithValues() keeps track of these associated values.

Warning

Really, this requires a group homomorphism from the abstract Abelian group to the set of values. This is only checked if you pass the check=True option to AbelianGroupWithValues().

EXAMPLES:

Here is $$\ZZ_6$$ with value $$-1$$ assigned to the generator:

sage: Z6 = AbelianGroupWithValues([-1], [6], names='g')
sage: g = Z6.gen(0)
sage: g.value()
-1
sage: g*g
g^2
sage: (g*g).value()
1
sage: for i in range(7):
....:     print((i, g^i, (g^i).value()))
(0, 1, 1)
(1, g, -1)
(2, g^2, 1)
(3, g^3, -1)
(4, g^4, 1)
(5, g^5, -1)
(6, 1, 1)

>>> from sage.all import *
>>> Z6 = AbelianGroupWithValues([-Integer(1)], [Integer(6)], names='g')
>>> g = Z6.gen(Integer(0))
>>> g.value()
-1
>>> g*g
g^2
>>> (g*g).value()
1
>>> for i in range(Integer(7)):
...     print((i, g**i, (g**i).value()))
(0, 1, 1)
(1, g, -1)
(2, g^2, 1)
(3, g^3, -1)
(4, g^4, 1)
(5, g^5, -1)
(6, 1, 1)


The elements come with a coercion embedding into the values_group(), so you can use the group elements instead of the values:

sage: # needs sage.rings.number_field
sage: CF3.<zeta> = CyclotomicField(3)
sage: Z3.<g> = AbelianGroupWithValues([zeta], [3])
sage: Z3.values_group()
Cyclotomic Field of order 3 and degree 2
sage: g.value()
zeta
sage: CF3(g)
zeta
sage: g + zeta
2*zeta
sage: zeta + g
2*zeta

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> CF3 = CyclotomicField(Integer(3), names=('zeta',)); (zeta,) = CF3._first_ngens(1)
>>> Z3 = AbelianGroupWithValues([zeta], [Integer(3)], names=('g',)); (g,) = Z3._first_ngens(1)
>>> Z3.values_group()
Cyclotomic Field of order 3 and degree 2
>>> g.value()
zeta
>>> CF3(g)
zeta
>>> g + zeta
2*zeta
>>> zeta + g
2*zeta

sage.groups.abelian_gps.values.AbelianGroupWithValues(values, n, gens_orders=None, names='f', check=False, values_group=None)[source]#

Construct an Abelian group with values associated to the generators.

INPUT:

• values – a list/tuple/iterable of values that you want to associate to the generators

• n – integer (optional). If not specified, will be derived

from gens_orders

• gens_orders – a list of non-negative integers in the form

$$[a_0, a_1, \dots, a_{n-1}]$$, typically written in increasing order. This list is padded with zeros if it has length less than n. The orders of the commuting generators, with $$0$$ denoting an infinite cyclic factor.

• names – (optional) names of generators

• values_group – a parent or None (default). The common parent of the values. This might be a group, but can also just contain the values. For example, if the values are units in a ring then the values_group would be the whole ring. If None it will be derived from the values.

EXAMPLES:

sage: G = AbelianGroupWithValues([-1], [6])
sage: g = G.gen(0)
sage: for i in range(7):
....:     print((i, g^i, (g^i).value()))
(0, 1, 1)
(1, f, -1)
(2, f^2, 1)
(3, f^3, -1)
(4, f^4, 1)
(5, f^5, -1)
(6, 1, 1)
sage: G.values_group()
Integer Ring

>>> from sage.all import *
>>> G = AbelianGroupWithValues([-Integer(1)], [Integer(6)])
>>> g = G.gen(Integer(0))
>>> for i in range(Integer(7)):
...     print((i, g**i, (g**i).value()))
(0, 1, 1)
(1, f, -1)
(2, f^2, 1)
(3, f^3, -1)
(4, f^4, 1)
(5, f^5, -1)
(6, 1, 1)
>>> G.values_group()
Integer Ring


The group elements come with a coercion embedding into the values_group(), so you can use them like their value()

sage: G.values_embedding()
Generic morphism:
From: Multiplicative Abelian group isomorphic to C6
To:   Integer Ring
sage: g.value()
-1
sage: 0 + g
-1
sage: 1 + 2*g
-1

>>> from sage.all import *
>>> G.values_embedding()
Generic morphism:
From: Multiplicative Abelian group isomorphic to C6
To:   Integer Ring
>>> g.value()
-1
>>> Integer(0) + g
-1
>>> Integer(1) + Integer(2)*g
-1

class sage.groups.abelian_gps.values.AbelianGroupWithValuesElement(parent, exponents, value=None)[source]#

An element of an Abelian group with values assigned to generators.

INPUT:

• exponents – tuple of integers; the exponent vector defining the group element

• parent – the parent

• value – the value assigned to the group element or None (default); in the latter case, the value is computed as needed

EXAMPLES:

sage: F = AbelianGroupWithValues([1,-1], [2,4])
sage: a,b = F.gens()
sage: TestSuite(a*b).run()

>>> from sage.all import *
>>> F = AbelianGroupWithValues([Integer(1),-Integer(1)], [Integer(2),Integer(4)])
>>> a,b = F.gens()
>>> TestSuite(a*b).run()

value()[source]#

Return the value of the group element.

OUTPUT: the value according to the values for generators; see gens_values()

EXAMPLES:

sage: G = AbelianGroupWithValues([5], 1)
sage: G.0.value()
5

>>> from sage.all import *
>>> G = AbelianGroupWithValues([Integer(5)], Integer(1))
>>> G.gen(0).value()
5

class sage.groups.abelian_gps.values.AbelianGroupWithValuesEmbedding(domain, codomain)[source]#

Bases: Morphism

The morphism embedding the Abelian group with values in its values group.

INPUT:

EXAMPLES:

sage: # needs sage.symbolic
sage: Z4.<g> = AbelianGroupWithValues([I], [4])
sage: embedding = Z4.values_embedding();  embedding
Generic morphism:
From: Multiplicative Abelian group isomorphic to C4
To:   Number Field in I with defining polynomial x^2 + 1 with I = 1*I
sage: embedding(1)
1
sage: embedding(g)
I
sage: embedding(g^2)
-1

>>> from sage.all import *
>>> # needs sage.symbolic
>>> Z4 = AbelianGroupWithValues([I], [Integer(4)], names=('g',)); (g,) = Z4._first_ngens(1)
>>> embedding = Z4.values_embedding();  embedding
Generic morphism:
From: Multiplicative Abelian group isomorphic to C4
To:   Number Field in I with defining polynomial x^2 + 1 with I = 1*I
>>> embedding(Integer(1))
1
>>> embedding(g)
I
>>> embedding(g**Integer(2))
-1

class sage.groups.abelian_gps.values.AbelianGroupWithValues_class(generator_orders, names, values, values_group)[source]#

The class of an Abelian group with values associated to the generator.

INPUT:

• generator_orders – tuple of integers; the orders of the generators.

• names – string or list of strings; the names for the generators

• values – tuple the same length as the number of generators; the values assigned to the generators.

• values_group – the common parent of the values

EXAMPLES:

sage: G.<a,b> = AbelianGroupWithValues([2,-1], [0,4])
sage: TestSuite(G).run()

>>> from sage.all import *
>>> G = AbelianGroupWithValues([Integer(2),-Integer(1)], [Integer(0),Integer(4)], names=('a', 'b',)); (a, b,) = G._first_ngens(2)
>>> TestSuite(G).run()

Element[source]#
gen(i=0)[source]#

The $$i$$-th generator of the abelian group.

INPUT:

• i – integer (default: 0); the index of the generator

OUTPUT: a group element

EXAMPLES:

sage: F = AbelianGroupWithValues([1,2,3,4,5], 5, [], names='a')
sage: F.0
a0
sage: F.0.value()
1
sage: F.2
a2
sage: F.2.value()
3

sage: G = AbelianGroupWithValues([-1,0,1], [2,1,3])
sage: G.gens()
(f0, 1, f2)

>>> from sage.all import *
>>> F = AbelianGroupWithValues([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)], Integer(5), [], names='a')
>>> F.gen(0)
a0
>>> F.gen(0).value()
1
>>> F.gen(2)
a2
>>> F.gen(2).value()
3

>>> G = AbelianGroupWithValues([-Integer(1),Integer(0),Integer(1)], [Integer(2),Integer(1),Integer(3)])
>>> G.gens()
(f0, 1, f2)

gens_values()[source]#

Return the values associated to the generators.

OUTPUT: tuple

EXAMPLES:

sage: G = AbelianGroupWithValues([-1,0,1], [2,1,3])
sage: G.gens()
(f0, 1, f2)
sage: G.gens_values()
(-1, 0, 1)

>>> from sage.all import *
>>> G = AbelianGroupWithValues([-Integer(1),Integer(0),Integer(1)], [Integer(2),Integer(1),Integer(3)])
>>> G.gens()
(f0, 1, f2)
>>> G.gens_values()
(-1, 0, 1)

values_embedding()[source]#

Return the embedding of self in values_group().

OUTPUT: a morphism

EXAMPLES:

sage: Z4 = AbelianGroupWithValues([I], [4])                                 # needs sage.symbolic
sage: Z4.values_embedding()                                                 # needs sage.symbolic
Generic morphism:
From: Multiplicative Abelian group isomorphic to C4
To:   Number Field in I with defining polynomial x^2 + 1 with I = 1*I

>>> from sage.all import *
>>> Z4 = AbelianGroupWithValues([I], [Integer(4)])                                 # needs sage.symbolic
>>> Z4.values_embedding()                                                 # needs sage.symbolic
Generic morphism:
From: Multiplicative Abelian group isomorphic to C4
To:   Number Field in I with defining polynomial x^2 + 1 with I = 1*I

values_group()[source]#

The common parent of the values.

The values need to form a multiplicative group, but can be embedded in a larger structure. For example, if the values are units in a ring then the values_group() would be the whole ring.

OUTPUT: the common parent of the values, containing the group generated by all values

EXAMPLES:

sage: G = AbelianGroupWithValues([-1,0,1], [2,1,3])
sage: G.values_group()
Integer Ring

sage: Z4 = AbelianGroupWithValues([I], [4])                                 # needs sage.symbolic
sage: Z4.values_group()                                                     # needs sage.symbolic
Number Field in I with defining polynomial x^2 + 1 with I = 1*I

>>> from sage.all import *
>>> G = AbelianGroupWithValues([-Integer(1),Integer(0),Integer(1)], [Integer(2),Integer(1),Integer(3)])
>>> G.values_group()
Integer Ring

>>> Z4 = AbelianGroupWithValues([I], [Integer(4)])                                 # needs sage.symbolic
>>> Z4.values_group()                                                     # needs sage.symbolic
Number Field in I with defining polynomial x^2 + 1 with I = 1*I