Value groups of discrete valuations

This file defines additive sub(semi-)groups of \(\QQ\) and related structures.

AUTHORS:

  • Julian Rüth (2013-09-06): initial version

EXAMPLES:

sage: v = ZZ.valuation(2)
sage: v.value_group()
Additive Abelian Group generated by 1
sage: v.value_semigroup()
Additive Abelian Semigroup generated by 1

>>> from sage.all import *
>>> v = ZZ.valuation(Integer(2))
>>> v.value_group()
Additive Abelian Group generated by 1
>>> v.value_semigroup()
Additive Abelian Semigroup generated by 1
class sage.rings.valuation.value_group.DiscreteValuationCodomain[source]

Bases: UniqueRepresentation, Parent

The codomain of discrete valuations, the rational numbers extended by \(\pm\infty\).

EXAMPLES:

sage: from sage.rings.valuation.value_group import DiscreteValuationCodomain
sage: C = DiscreteValuationCodomain(); C
Codomain of Discrete Valuations

>>> from sage.all import *
>>> from sage.rings.valuation.value_group import DiscreteValuationCodomain
>>> C = DiscreteValuationCodomain(); C
Codomain of Discrete Valuations
class sage.rings.valuation.value_group.DiscreteValueGroup(generator)[source]

Bases: UniqueRepresentation, Parent

The value group of a discrete valuation, an additive subgroup of \(\QQ\) generated by generator.

INPUT:

  • generator – a rational number

Note

We do not rely on the functionality provided by additive abelian groups in Sage since these require the underlying set to be the integers. Therefore, we roll our own Z-module here. We could have used AdditiveAbelianGroupWrapper here, but it seems to be somewhat outdated. In particular, generic group functionality should now come from the category and not from the super-class. A facade of Q appeared to be the better approach.

EXAMPLES:

sage: from sage.rings.valuation.value_group import DiscreteValueGroup
sage: D1 = DiscreteValueGroup(0); D1
Trivial Additive Abelian Group
sage: D2 = DiscreteValueGroup(4/3); D2
Additive Abelian Group generated by 4/3
sage: D3 = DiscreteValueGroup(-1/3); D3
Additive Abelian Group generated by 1/3

>>> from sage.all import *
>>> from sage.rings.valuation.value_group import DiscreteValueGroup
>>> D1 = DiscreteValueGroup(Integer(0)); D1
Trivial Additive Abelian Group
>>> D2 = DiscreteValueGroup(Integer(4)/Integer(3)); D2
Additive Abelian Group generated by 4/3
>>> D3 = DiscreteValueGroup(-Integer(1)/Integer(3)); D3
Additive Abelian Group generated by 1/3
denominator()[source]

Return the denominator of a generator of this group.

EXAMPLES:

sage: from sage.rings.valuation.value_group import DiscreteValueGroup
sage: DiscreteValueGroup(3/8).denominator()
8

>>> from sage.all import *
>>> from sage.rings.valuation.value_group import DiscreteValueGroup
>>> DiscreteValueGroup(Integer(3)/Integer(8)).denominator()
8
gen()[source]

Return a generator of this group.

EXAMPLES:

sage: from sage.rings.valuation.value_group import DiscreteValueGroup
sage: DiscreteValueGroup(-3/8).gen()
3/8

>>> from sage.all import *
>>> from sage.rings.valuation.value_group import DiscreteValueGroup
>>> DiscreteValueGroup(-Integer(3)/Integer(8)).gen()
3/8
index(other)[source]

Return the index of other in this group.

INPUT:

  • other – a subgroup of this group

EXAMPLES:

sage: from sage.rings.valuation.value_group import DiscreteValueGroup
sage: DiscreteValueGroup(3/8).index(DiscreteValueGroup(3))
8
sage: DiscreteValueGroup(3).index(DiscreteValueGroup(3/8))
Traceback (most recent call last):
...
ValueError: other must be a subgroup of this group
sage: DiscreteValueGroup(3).index(DiscreteValueGroup(0))
Traceback (most recent call last):
...
ValueError: other must have finite index in this group
sage: DiscreteValueGroup(0).index(DiscreteValueGroup(0))
1
sage: DiscreteValueGroup(0).index(DiscreteValueGroup(3))
Traceback (most recent call last):
...
ValueError: other must be a subgroup of this group

>>> from sage.all import *
>>> from sage.rings.valuation.value_group import DiscreteValueGroup
>>> DiscreteValueGroup(Integer(3)/Integer(8)).index(DiscreteValueGroup(Integer(3)))
8
>>> DiscreteValueGroup(Integer(3)).index(DiscreteValueGroup(Integer(3)/Integer(8)))
Traceback (most recent call last):
...
ValueError: other must be a subgroup of this group
>>> DiscreteValueGroup(Integer(3)).index(DiscreteValueGroup(Integer(0)))
Traceback (most recent call last):
...
ValueError: other must have finite index in this group
>>> DiscreteValueGroup(Integer(0)).index(DiscreteValueGroup(Integer(0)))
1
>>> DiscreteValueGroup(Integer(0)).index(DiscreteValueGroup(Integer(3)))
Traceback (most recent call last):
...
ValueError: other must be a subgroup of this group
is_trivial()[source]

Return whether this is the trivial additive abelian group.

EXAMPLES:

sage: from sage.rings.valuation.value_group import DiscreteValueGroup
sage: DiscreteValueGroup(-3/8).is_trivial()
False
sage: DiscreteValueGroup(0).is_trivial()
True

>>> from sage.all import *
>>> from sage.rings.valuation.value_group import DiscreteValueGroup
>>> DiscreteValueGroup(-Integer(3)/Integer(8)).is_trivial()
False
>>> DiscreteValueGroup(Integer(0)).is_trivial()
True
numerator()[source]

Return the numerator of a generator of this group.

EXAMPLES:

sage: from sage.rings.valuation.value_group import DiscreteValueGroup
sage: DiscreteValueGroup(3/8).numerator()
3

>>> from sage.all import *
>>> from sage.rings.valuation.value_group import DiscreteValueGroup
>>> DiscreteValueGroup(Integer(3)/Integer(8)).numerator()
3
some_elements()[source]

Return some typical elements in this group.

EXAMPLES:

sage: from sage.rings.valuation.value_group import DiscreteValueGroup
sage: DiscreteValueGroup(-3/8).some_elements()
[3/8, -3/8, 0, 42, 3/2, -3/2, 9/8, -9/8]

>>> from sage.all import *
>>> from sage.rings.valuation.value_group import DiscreteValueGroup
>>> DiscreteValueGroup(-Integer(3)/Integer(8)).some_elements()
[3/8, -3/8, 0, 42, 3/2, -3/2, 9/8, -9/8]
class sage.rings.valuation.value_group.DiscreteValueSemigroup(generators)[source]

Bases: UniqueRepresentation, Parent

The value semigroup of a discrete valuation, an additive subsemigroup of \(\QQ\) generated by generators.

INPUT:

  • generators – rational numbers

EXAMPLES:

sage: from sage.rings.valuation.value_group import DiscreteValueSemigroup
sage: D1 = DiscreteValueSemigroup(0); D1
Trivial Additive Abelian Semigroup
sage: D2 = DiscreteValueSemigroup(4/3); D2
Additive Abelian Semigroup generated by 4/3
sage: D3 = DiscreteValueSemigroup([-1/3, 1/2]); D3
Additive Abelian Semigroup generated by -1/3, 1/2

>>> from sage.all import *
>>> from sage.rings.valuation.value_group import DiscreteValueSemigroup
>>> D1 = DiscreteValueSemigroup(Integer(0)); D1
Trivial Additive Abelian Semigroup
>>> D2 = DiscreteValueSemigroup(Integer(4)/Integer(3)); D2
Additive Abelian Semigroup generated by 4/3
>>> D3 = DiscreteValueSemigroup([-Integer(1)/Integer(3), Integer(1)/Integer(2)]); D3
Additive Abelian Semigroup generated by -1/3, 1/2
gens()[source]

Return the generators of this semigroup.

EXAMPLES:

sage: from sage.rings.valuation.value_group import DiscreteValueSemigroup
sage: DiscreteValueSemigroup(-3/8).gens()
(-3/8,)

>>> from sage.all import *
>>> from sage.rings.valuation.value_group import DiscreteValueSemigroup
>>> DiscreteValueSemigroup(-Integer(3)/Integer(8)).gens()
(-3/8,)
is_group()[source]

Return whether this semigroup is a group.

EXAMPLES:

sage: from sage.rings.valuation.value_group import DiscreteValueSemigroup
sage: DiscreteValueSemigroup(1).is_group()
False
sage: D = DiscreteValueSemigroup([-1, 1])
sage: D.is_group()
True

>>> from sage.all import *
>>> from sage.rings.valuation.value_group import DiscreteValueSemigroup
>>> DiscreteValueSemigroup(Integer(1)).is_group()
False
>>> D = DiscreteValueSemigroup([-Integer(1), Integer(1)])
>>> D.is_group()
True

Invoking this method also changes the category of this semigroup if it is a group:

sage: D in AdditiveMagmas().AdditiveAssociative().AdditiveUnital().AdditiveInverse()
True

>>> from sage.all import *
>>> D in AdditiveMagmas().AdditiveAssociative().AdditiveUnital().AdditiveInverse()
True
is_trivial()[source]

Return whether this is the trivial additive abelian semigroup.

EXAMPLES:

sage: from sage.rings.valuation.value_group import DiscreteValueSemigroup
sage: DiscreteValueSemigroup(-3/8).is_trivial()
False
sage: DiscreteValueSemigroup([]).is_trivial()
True

>>> from sage.all import *
>>> from sage.rings.valuation.value_group import DiscreteValueSemigroup
>>> DiscreteValueSemigroup(-Integer(3)/Integer(8)).is_trivial()
False
>>> DiscreteValueSemigroup([]).is_trivial()
True
some_elements()[source]

Return some typical elements in this semigroup.

EXAMPLES:

sage: from sage.rings.valuation.value_group import DiscreteValueSemigroup
sage: list(DiscreteValueSemigroup([-3/8,1/2]).some_elements())              # needs sage.numerical.mip
[0, -3/8, 1/2, ...]

>>> from sage.all import *
>>> from sage.rings.valuation.value_group import DiscreteValueSemigroup
>>> list(DiscreteValueSemigroup([-Integer(3)/Integer(8),Integer(1)/Integer(2)]).some_elements())              # needs sage.numerical.mip
[0, -3/8, 1/2, ...]