abelian group

Abelian Group

Mathematically an Abelian Group is a Commutative Monoid G all of whose elements have additive inverses.

Note

Addition is used for the group operation.

Important

Contract: AbelianGroup initializer parameters must have

  • add closed, associative and commutative on reps

  • zero additive identity on reps, rep.add(zero) == rep == zero.add(rep)

  • negate must me idempotent: neg(neg(rep)) == rep

class boring_math.abstract_algebra.algebras.abelian_group.AbelianGroup(add: ~collections.abc.Callable[[H, H], H], zero: H, negate: ~collections.abc.Callable[[H], H], process: ~collections.abc.Callable[[H], H] = <function AbelianGroup.<lambda>>)
Parameters:

  • add – Closed, commutative and associative function on reps.

  • zero – Representation for additive identity.

  • negate – Function mapping element representation to the representation of corresponding negated element.

__init__(add: ~collections.abc.Callable[[H, H], H], zero: H, negate: ~collections.abc.Callable[[H], H], process: ~collections.abc.Callable[[H], H] = <function AbelianGroup.<lambda>>)
Parameters:

  • add – Closed, commutative and associative function on reps.

  • zero – Representation for additive identity.

  • negate – Function mapping element representation to the representation of corresponding negated element.

__call__(rep: H) → AbelianGroupElement

Add the unique element to the abelian group with a given rep.

Parameters:

rep – Representation to add if not already present.

Returns:

The unique element with that representation.

class boring_math.abstract_algebra.algebras.abelian_group.AbelianGroupElement(rep: H, algebra: AbelianGroup[H], process: ~collections.abc.Callable[[H], H] = <function AbelianGroupElement.<lambda>>)
__init__(rep: H, algebra: AbelianGroup[H], process: ~collections.abc.Callable[[H], H] = <function AbelianGroupElement.<lambda>>) → None
__mul__(n: Self | int) → Self

Multiplying additive group element by an integer n>=0 is the same as repeated addition.

Parameters:

n – Add abelian group element to itself n >= 0 times.

Returns:

The sum of the group element n times.

Raises:

  • TypeError – if given an element instead of an int.

  • ValueError – If add method was not defined on the group.

__neg__() → Self

Negate the abelian group element.

Returns:

The unique additive inverse element to self.

Raises:

ValueError – If algebra fails to have additive inverses.