commutative monoid

Commutative Monoid

Mathematically a commutative Monoid is a Semigroup M along with an identity element u, that is (∃u ∈ M) => (∀m ∈ M)(u+m = m+u = m).

When such an identity element u exists, it is necessarily unique.

Important

Contract: Commutative Monoid initializer parameters must have

  • add closed commutative and associative on reps

  • zero an identity on reps, rep+zero == rep == zero+rep

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

  • add – Closed commutative and associative function reps.

  • zero – Representation for additive identity.

Returns:

A commutative monoid algebra.

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

  • add – Closed commutative and associative function reps.

  • zero – Representation for additive identity.

Returns:

A commutative monoid algebra.

__call__(rep: H) → CommutativeMonoidElement

Add the unique element to the monoid 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.commutative_monoid.CommutativeMonoidElement(rep: H, algebra: CommutativeMonoid[H])
__init__(rep: H, algebra: CommutativeMonoid[H]) → None
__str__() → str

Return str(self).

__mul__(n: int | Self) → Self

Repeatedly add a commutative monoid element n>=0 times.

Parameters:

n – The number of times to add the element to itself.

Returns:

The element added to its additive identity n times.

Raises:

  • TypeError – If self and other are different types.

  • ValueError – If self and other are same type but different concrete algebras.

  • ValueError – If algebra fails to have an additive identity element.