monoid¶

Monoid¶

Monoid

Mathematically a Monoid is a Semigroup M such that

(∃u ∈ M) => (∀m ∈ M)(u*m = m*u = m)

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

Important

Contract: Monoid initializer parameters must have

class boring_math.abstract_algebra.algebras.monoid.Monoid¶

Bases: Semigroup, Generic

Parameters:

mult – Associative function H X H -> H on representations.
one – Representation for multiplicative identity.
narrow – Narrow the rep type, many-to-one function. Like choosing an element from a coset of a group.

__init__(mult: Callable[[H, H], H], one: H, narrow: Callable[[H], H] | None = None)¶

Parameters:

mult – Associative function H X H -> H on representations.
one – Representation for multiplicative identity.
narrow – Narrow the rep type, many-to-one function. Like choosing an element from a coset of a group.

class boring_math.abstract_algebra.algebras.monoid.MonoidElement¶

__str__() → str¶

__pow__(n: int) → Self¶

Raise the element to power to the power of n>=0.

Parameters:

n – The int power to raise the element to.

Returns:

The element (or its inverse) raised to the integer n power.

Raises:

TypeError – If self and other are different types.
ValueError – If self and other are same type but different concrete groups.
ValueError – If algebra fails to have an identity element.

__mul__(right: object) → Self¶

Multiply two elements of the same concrete algebra together.

Parameters:: right – An element within the same concrete algebra or a right action.
Returns:: The product self * right otherwise NotImplemented.
Raises:: ValueError – If self and right are same type but different concrete algebras.

__rmul__(left: object) → Self¶

When left side of multiplication does not know how to multiply right side.