A commutative associative binary operation with an identity.

Rationale:

• This is the common superclass of AbelianGroup and Semiring. General monoids are also useful, but it is a common expectation that an operation denoted by + is commutative.
• zero is required because this class is insufficient for integer literals.
• Ideally, 0 could be defined as equivalent to zero, with other integer literals handled by fromNatural.

Faster implementation of atimes when addition is idempotent.

Faster implementation of atimes or gtimes when x+x = zero.

An Abelian group has a commutative associative binary operation with an identity and inverses.

Rationale:

• The abs and signum operations lack sensible definitions for many useful instances, such as complex numbers, polynomials, matrices, etc.
• Types that have subtraction but not multiplication include vectors and dimensioned quantities.

The same as flip (-).

Because - is treated specially in the Haskell grammar, (- e) is not a section, but an application of prefix negation. However, (subtract exp) is equivalent to the disallowed section.

Faster implementation of gtimes when addition is idempotent.

A semiring: addition defines a commutative monoid, and multiplication defines a monoid and distributes over addition and zero. Multiplication is not guaranteed to be commutative.

Rationale:

• Natural is the key example of a type with multiplication but not subtraction, but there are many more.
• one is required because this class is insufficient for integer literals.
• In an ideal world, an integer literal i would be treated as equivalent to fromNatural i and have type (Semiring a) => a. (The lexical syntax already permits only non-negative numbers.)
• rescale is available here with a trivial default definition so that some operations on complex numbers, whose direct definitions would often overflow on Float or Double components, can be defined for other types as well.

A ring: addition forms an Abelian group, and multiplication defines a monoid and distributes over addition. Multiplication is not guaranteed to be commutative.

Rationale:

• This is sufficient to define fromInteger, but often a much more efficient definition is available.

A cancellative commutative semiring with a designated canonical factoring of each value as the product of a unit (invertible value) and an associate.

Rationale:

• This normalization is required so that the results of operations such as gcd and lcm can be uniquely defined. In the original Prelude, abs and signum were used for this, but they lack coherent definitions on many useful instances.
• This class is usually used together with Euclidean, and generally has matching instances. Merging the two classes would be a possibility. That would allow a default definition of stdAssociate.

A Euclidean semiring: a commutative semiring with Euclidean division, yielding a quotient and a remainder that is smaller than the divisor in a well-founded ordering. This is sufficient to implement Euclid's algorithm for the greatest common divisor.

Rationale:

• This class, together with StandardAssociate, is sufficient to define gcd, and thus to define arithmetic operations on Ratio a.
• The uniformity condition is required to make modular arithmetic work. Non-integer examples include Complex Integer (Gaussian integers) and polynomials.
• The usual definition of a Euclidean domain assumes a ring, but division with remainder can be defined in the absence of negation, e.g. for Natural.

gcd x y is a common factor of x and y such that

• stdAssociate (gcd x y) = gcd x y, and
• any common factor of x and y is a factor of gcd x y.

lcm x y is a common multiple of x and y such that

• stdAssociate (lcm x y) = lcm x y, and
• any common multiple of x and y is a multiple of lcm x y.

bezout x y = (a, b) such that a*x + b*y = gcd x y (Bézout's identity).

In particular, if x and y are coprime (i.e. gcd x y == one),

• b is the multiplicative inverse of y modulo x.
• a is the multiplicative inverse of x modulo y.
• j*a*x + i*b*y is equivalent to i modulo x and to j modulo y (Chinese Remainder Theorem).

The list of quadruples \((q_i, r_i, s_i, t_i)\) generated by the extended Euclidean algorithm, which is a maximal list satisfying:

• \(r_{i-1} = q_i r_i + r_{i+1}\) with \(r_{i+1}\) smaller than \(r_i\), where \(r_0 = a\) and \(r_1 = b\), and
• \(r_i = s_i a + t_i b\).

The last \(r_i\) in the list is a greatest common divisor of \(a\) and \(b\), so that the second equation above becomes Bézout's identity.

A Semiring in which all non-zero elements have multiplicative inverses.

Rationale:

• Quaternions have multiplicative inverses, but do not form a field, because multiplication is not commutative.
• Some semirings, such as the tropical semiring and its dual, support division but not subtraction.

A commutative Semiring in which all non-zero elements have multiplicative inverses, so that division is uniquely defined.

Rationale:

• Some semirings, such as the tropical semiring and its dual, support division but not subtraction.

A Ring in which all non-zero elements have multiplicative inverses.

Rationale:

• Quaternions have multiplicative inverses, but multiplication is not commutative, so they do not have a well-defined division operation, and do not form a field.

A commutative Ring in which all non-zero elements have multiplicative inverses.

Rationale:

• fromRational needs to be in an independent class, because Rational cannot be embedded in finite fields.
• While fields trivially support Euclidean division and standard associates, they are kept apart for backward compatibility.

Rings extending Rational. Such a ring should have characteristic 0, i.e. no sum 1 + ... + 1 should equal zero.

Rationale:

• For fromRational to be injective, its domain must be infinite, which naturally excludes finite fields. Indeed Rational is initial in the category of infinite fields. (It's true that Float and Double aren't infinite, but they aren't even additive monoids.)
• Conversely, many rings that are not fields support embedding of rationals, e.g. polynomials and matrices.

Types that can be faithfully embedded in Rational.

Rationale:

• This is essentially equivalent to the old Real class, but with the Num superclass reduced to Semiring (so it does not assume negative values).

Types representing a contiguous set of integers, including 0, 1 and 2.

Rationale:

• This is similar to the old Integral class, but does not require subtraction, which does not work for Natural.

A differential semiring

A differential semiring with anti-differentiation

A functor on additive monoids