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

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

Python

>>> 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

Sage Live

v = ZZ.valuation(2)
v.value_group()
v.value_semigroup()

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

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

Python

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

Sage Live

from sage.rings.valuation.value_group import DiscreteValuationCodomain
C = DiscreteValuationCodomain(); C

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 We could have used AdditiveAbelianGroupWrapper here, but it seems to be somewhat outdated.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

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

Python

>>> 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

Sage Live

from sage.rings.valuation.value_group import DiscreteValueGroup
D1 = DiscreteValueGroup(0); D1
D2 = DiscreteValueGroup(4/3); D2
D3 = DiscreteValueGroup(-1/3); D3

denominator()[source]¶

Return the denominator of a generator of this group.

EXAMPLES:

Sage

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

Python

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

Sage Live

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

gen()[source]¶

Return a generator of this group.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.rings.valuation.value_group import DiscreteValueGroup
>>> DiscreteValueGroup(-Integer(3)/Integer(8)).gen()
3/8

Sage Live

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

index(other)[source]¶

Return the index of other in this group.

INPUT:

other – a subgroup of this group

EXAMPLES:

Sage

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

Python

>>> 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

Sage Live

from sage.rings.valuation.value_group import DiscreteValueGroup
DiscreteValueGroup(3/8).index(DiscreteValueGroup(3))
DiscreteValueGroup(3).index(DiscreteValueGroup(3/8))
DiscreteValueGroup(3).index(DiscreteValueGroup(0))
DiscreteValueGroup(0).index(DiscreteValueGroup(0))
DiscreteValueGroup(0).index(DiscreteValueGroup(3))

is_trivial()[source]¶

Return whether this is the trivial additive abelian group.

EXAMPLES:

Sage

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

Python

>>> 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

Sage Live

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

numerator()[source]¶

Return the numerator of a generator of this group.

EXAMPLES:

Sage

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

Python

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

Sage Live

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

some_elements()[source]¶

Return some typical elements in this group.

EXAMPLES:

Sage

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]

Python

>>> 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]

Sage Live

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

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

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

Python

>>> 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

Sage Live

from sage.rings.valuation.value_group import DiscreteValueSemigroup
D1 = DiscreteValueSemigroup(0); D1
D2 = DiscreteValueSemigroup(4/3); D2
D3 = DiscreteValueSemigroup([-1/3, 1/2]); D3

gens()[source]¶

Return the generators of this semigroup.

EXAMPLES:

Sage

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

Python

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

Sage Live

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

is_group()[source]¶

Return whether this semigroup is a group.

EXAMPLES:

Sage

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

Python

>>> 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

Sage Live

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

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

Sage

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

Python

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

Sage Live

D in AdditiveMagmas().AdditiveAssociative().AdditiveUnital().AdditiveInverse()

is_trivial()[source]¶

Return whether this is the trivial additive abelian semigroup.

EXAMPLES:

Sage

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

Python

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

Sage Live

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

some_elements()[source]¶

Return some typical elements in this semigroup.

EXAMPLES:

Sage

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, ...]

Python

>>> 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, ...]

Sage Live

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