ring

Ring

Mathematically a Ring is an abelian group under addition and a Monoid under multiplication. The additive and multiplicative identities are denoted one and zero respectfully.

By convention one != zero, otherwise the algebra consists of just one unique element.

Important

Contract: Ring initializer parameters must have

• add closed, commutative and associative on reps

• mult closed and associative on reps

• one an identity on reps, rep*one == rep == one*rep

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

• negate maps rep -> -rep, rep + negate(rep) == zero

• zero != one

class boring_math.abstract_algebra.algebras.ring.Ring(add: ~collections.abc.Callable[[H, H], H], mult: ~collections.abc.Callable[[H, H], H], one: H, zero: H, negate: ~collections.abc.Callable[[H], H], process: ~collections.abc.Callable[[H], H] = <function Ring.<lambda>>)

Parameters:

• add – Closed commutative and associative function reps.

• mult – Closed associative function reps.

• one – Representation for multiplicative identity.

• 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], mult: ~collections.abc.Callable[[H, H], H], one: H, zero: H, negate: ~collections.abc.Callable[[H], H], process: ~collections.abc.Callable[[H], H] = <function Ring.<lambda>>)

Parameters:

• add – Closed commutative and associative function reps.

• mult – Closed associative function reps.

• one – Representation for multiplicative identity.

• zero – Representation for additive identity.

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

__call__(rep: H) → RingElement

Add the unique element to the ring 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.ring.RingElement(rep: H, algebra: Ring[H])

__init__(rep: H, algebra: Ring[H]) → None

__str__() → str

Return str(self).

__mul__(other: object) → Self

Multiplication * operator.

• Multiplying element by an integer n>=0 is repeated addition.

• Algebra mult if other is a member of the same concrete algebra.

• Otherwise return NotImplemented (for a right action)

Parameters:

other – Add element to itself n >= 0 times.

Returns:

The sum of the element n times.

Raises:

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

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

__rmul__(other: object) → Self

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

• Multiplying element by an integer n>=0 is repeated addition.

• If other not member of same concrete algebra or left mult would of worked.

• Otherwise return NotImplemented (for a left action)

__pow__(n: int) → Self

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

Parameters:

n – The int power to raise the element to.

Returns:

The element raised to a non-negative int power.

Raises:

• ValueError – If algebra is not multiplicative.

• ValueError – If algebra does not have a multiplicative identity element.

• ValueError – If n < 0.