or Addition

For practical reasons, we begin the class tree with Additive. Starting with Associative and Unital, or using Semigroup and Monoid from base tends to confuse the interface once you start having to disinguish between (say) monoidal addition and monoidal multiplication.

\a -> zero + a == a

\a -> a + zero == a

\a b c -> (a + b) + c == a + (b + c)

\a b -> a + b == b + a

By convention, (+) is regarded as commutative, but this is not universal, and the introduction of another symbol which means non-commutative addition seems a bit dogmatic.

>>> zero + 1
1

>>> 1 + 1
2

Compute the sum of a Foldable.

>>> sum [0..10]
55

Compute the accumulating sum of a Traversable.

>>> accsum [0..10]
[0,1,3,6,10,15,21,28,36,45,55]

or Subtraction

\a -> a - a == zero

\a -> negate a == zero - a

\a -> negate a + a == zero

\a -> a + negate a == zero

>>> negate 1
-1

>>> 1 - 2
-1

or Multiplication

For practical reasons, we begin the class tree with Additive and Multiplicative. Starting with Associative and Unital, or using Semigroup and Monoid from base tends to confuse the interface once you start having to disinguish between (say) monoidal addition and monoidal multiplication.

\a -> one * a == a

\a -> a * one == a

\a b c -> (a * b) * c == a * (b * c)

By convention, (*) is regarded as not necessarily commutative, but this is not universal, and the introduction of another symbol which means commutative multiplication seems a bit dogmatic.

>>> one * 2
2

>>> 2 * 3
6

Compute the product of a Foldable.

>>> product [1..5]
120

Compute the accumulating product of a Traversable.

>>> accproduct [1..5]
[1,2,6,24,120]

or Division

Though unusual, the term Divisive usefully fits in with the grammer of other classes and avoids name clashes that occur with some popular libraries.

\(a :: Double) -> a / a ~= one || a == zero

\(a :: Double) -> recip a ~= one / a || a == zero

\(a :: Double) -> recip a * a ~= one || a == zero

\(a :: Double) -> a * recip a ~= one || a == zero

>>> recip 2.0
0.5

>>> 1 / 2
0.5

Distributive

\a b c -> a * (b + c) == a * b + a * c

\a b c -> (a + b) * c == a * c + b * c

\a -> zero * a == zero

\a -> a * zero == zero

The sneaking in of the Absorption laws here glosses over the possibility that the multiplicative zero element does not have to correspond with the additive unital zero.

A Ring is an abelian group under addition (Unital, Associative, Commutative, Invertible) and monoidal under multiplication (Unital, Associative), and where multiplication distributes over addition.

\a -> zero + a == a
\a -> a + zero == a
\a b c -> (a + b) + c == a + (b + c)
\a b -> a + b == b + a
\a -> a - a == zero
\a -> negate a == zero - a
\a -> negate a + a == zero
\a -> a + negate a == zero
\a -> one * a == a
\a -> a * one == a
\a b c -> (a * b) * c == a * (b * c)
\a b c -> a * (b + c) == a * b + a * c
\a b c -> (a + b) * c == a * c + b * c
\a -> zero * a == zero
\a -> a * zero == zero

A StarSemiring is a semiring with an additional unary operator (star) satisfying:

\a -> star a == one + a * star a

A Kleene Algebra is a Star Semiring with idempotent addition.

a * x + x = a ==> star a * x + x = x
x * a + x = a ==> x * star a + x = x

Involutive Ring

adj (a + b) ==> adj a + adj b
adj (a * b) ==> adj a * adj b
adj one ==> one
adj (adj a) ==> a

Note: elements for which adj a == a are called "self-adjoint".

Defining two requires adding the multiplicative unital to itself. In other words, the concept of two is a Ring one.

>>> two
2

A Field is a set on which addition, subtraction, multiplication, and division are defined. It is also assumed that multiplication is distributive over addition.

A summary of the rules inherited from super-classes of Field:

zero + a == a
a + zero == a
((a + b) + c) (a + (b + c))
a + b == b + a
a - a == zero
negate a == zero - a
negate a + a == zero
a + negate a == zero
one * a == a
a * one == a
((a * b) * c) == (a * (b * c))
(a * (b + c)) == (a * b + a * c)
((a + b) * c) == (a * c + b * c)
a * zero == zero
zero * a == zero
a / a == one || a == zero
recip a == one / a || a == zero
recip a * a == one || a == zero
a * recip a == one || a == zero

A hyperbolic field class

\a -> a < zero || (sqrt . (**2)) a == a

\a -> a < zero || (log . exp) a ~= a

\a b -> (b < zero) || a <= zero || a == 1 || abs (a ** logBase a b - b) < 10 * epsilon

Quotienting of a Field into a Ring

See Field of fractions

\a -> a - one < floor a <= a <= ceiling a < a + one

Trigonometric Field

The list of laws is quite long: trigonometric identities

infinity is defined for any Field.

>>> one / zero + infinity
Infinity

>>> infinity + 1
Infinity

negative infinity

>>> negInfinity + infinity
NaN

nan is defined as zero/zero

but note the (social) law:

>>> nan == zero / zero
False

A half is a Field because it requires addition, multiplication and division.

>>> half :: Double
0.5

A algebraic structure with element joins: See Semilattice

Associativity: x \/ (y \/ z) == (x \/ y) \/ z
Commutativity: x \/ y == y \/ x
Idempotency:   x \/ x == x

The partial ordering induced by the join-semilattice structure

The partial ordering induced by the join-semilattice structure

A algebraic structure with element meets: See Semilattice

Associativity: x /\ (y /\ z) == (x /\ y) /\ z
Commutativity: x /\ y == y /\ x
Idempotency:   x /\ x == x

The partial ordering induced by the meet-semilattice structure

The partial ordering induced by the meet-semilattice structure

A join-semilattice with an identity element bottom for \/.

Identity: x \/ bottom == x

A meet-semilattice with an identity element top for /\.

Identity: x /\ top == x

Additive Action

m |+ zero == m

flipped additive action

(+|) == flip (|+)
zero +| m = m

Subtractive Action

m |- zero = m

Subtraction with the scalar on the left

(-|) == (+|) . negate
zero -| m = negate m

Multiplicative Action

m |* one = m
m |* zero = zero

flipped multiplicative action

(*|) == flip (|*)
one *| m = one
zero *| m = zero

Divisive Action

m |/ one = m

left scalar division

(/|) == (*|) . recip
one |/ m = recip m

A Module

a *| one == a
(a + b) *| c == (a *| c) + (b *| c)
c |* (a + b) == (c |* a) + (c |* b)
a *| zero == zero
a *| b == b |* a

Basis encapsulates the notion of magnitude (intuitively the quotienting of a higher-kinded number to a scalar one) and the basis on which the magnitude quotienting was performed. An instance needs to satisfy these laws:

\a -> magnitude a >= zero
\a -> magnitude zero == zero
\a -> a == magnitude a *| basis a
\a -> magnitude (basis a) == one

The names chosen are meant to represent the spiritual idea of a basis rather than a specific mathematics. See https://en.wikipedia.org/wiki/Basis_(linear_algebra) & https://en.wikipedia.org/wiki/Norm_(mathematics) for some mathematical motivations.

>>> magnitude (-0.5 :: Double)
0.5

>>> basis (-0.5 :: Double)
-1.0

Basis where the domain and magnitude codomain are the same.

Basis where the domain and basis codomain are the same.

Basis where the domain, magnitude codomain and basis codomain are the same.

The absolute value of a number.

\a -> abs a * signum a ~= a

>>> abs (-1)
1

The sign of a number.

>>> signum (-1)
-1

abs zero == zero, so any value for signum zero is ok. We choose lawful neutral:

>>> signum zero == zero
True

Distance, which combines the Subtractive notion of difference, with Basis.

distance a b >= zero
distance a a == zero
distance a b *| basis (a - b) == a - b

Convert between a "co-ordinated" or "higher-kinded" number and a direction.

ray . angle == basis
magnitude (ray x) == one

Something that has a magnitude and a direction, with both expressed as the same type.

See Polar coordinate system

Convert a higher-kinded number that has direction, to a Polar

Convert a Polar to a (higher-kinded) number that has a direction.

A small number, especially useful for approximate equality.