License | MIT |
---|---|
Maintainer | mail@doisinkidney.com |
Stability | experimental |
Safe Haskell | None |
Language | Haskell2010 |
Documentation
class Semiring a where Source #
A Semiring is like the
the combination of two Monoid
s. The first
is called <+>
; it has the identity element zero
, and it is
commutative. The second is called <.>
; it has identity element one
,
and it must distribute over <+>
.
Laws
Normal Monoid
laws
(a
<+>
b)<+>
c = a<+>
(b<+>
c)zero
<+>
a = a<+>
zero
= a(a
<.>
b)<.>
c = a<.>
(b<.>
c)one
<.>
a = a<.>
one
= a
Commutativity of <+>
Distribution of <.>
over <+>
Annihilation
The identity of <+>
.
The identity of <.>
.
(<.>) :: a -> a -> a infixl 7 Source #
An associative binary operation, which distributes over <+>
.
(<+>) :: a -> a -> a infixl 6 Source #
An associative, commutative binary operation.
The identity of <+>
.
The identity of <.>
.
(<+>) :: Num a => a -> a -> a infixl 6 Source #
An associative, commutative binary operation.
(<.>) :: Num a => a -> a -> a infixl 7 Source #
An associative binary operation, which distributes over <+>
.
Monad Add Source # | |
Functor Add Source # | |
Applicative Add Source # | |
Foldable Add Source # | |
Generic1 Add Source # | |
Bounded a => Bounded (Add a) Source # | |
Enum a => Enum (Add a) Source # | |
Eq a => Eq (Add a) Source # | |
Num a => Num (Add a) Source # | |
Ord a => Ord (Add a) Source # | |
Read a => Read (Add a) Source # | |
Show a => Show (Add a) Source # | |
Generic (Add a) Source # | |
Semiring a => Semigroup (Add a) Source # | |
Semiring a => Monoid (Add a) Source # | |
Semiring a => Semiring (Add a) Source # | |
type Rep1 Add Source # | |
type Rep (Add a) Source # | |
Monad Mul Source # | |
Functor Mul Source # | |
Applicative Mul Source # | |
Foldable Mul Source # | |
Generic1 Mul Source # | |
Bounded a => Bounded (Mul a) Source # | |
Enum a => Enum (Mul a) Source # | |
Eq a => Eq (Mul a) Source # | |
Num a => Num (Mul a) Source # | |
Ord a => Ord (Mul a) Source # | |
Read a => Read (Mul a) Source # | |
Show a => Show (Mul a) Source # | |
Generic (Mul a) Source # | |
Semiring a => Semigroup (Mul a) Source # | |
Semiring a => Monoid (Mul a) Source # | |
Semiring a => Semiring (Mul a) Source # | |
type Rep1 Mul Source # | |
type Rep (Mul a) Source # | |
The "Arctic" or max-plus semiring. It is a semiring where:
<+>
=max
zero
= -∞ -- represented byNothing
<.>
=<+>
one
=zero
Note that we can't use Max
from Semigroup
because annihilation needs to hold:
-∞<+>
x = x<+>
-∞ = -∞
Taking -∞ to be minBound
would break the above law. Using Nothing
to represent it follows the law.
Monad Max Source # | |
Functor Max Source # | |
Applicative Max Source # | |
Foldable Max Source # | |
Generic1 Max Source # | |
Eq a => Eq (Max a) Source # | |
Ord a => Ord (Max a) Source # | |
Read a => Read (Max a) Source # | |
Show a => Show (Max a) Source # | |
Generic (Max a) Source # | |
Ord a => Semigroup (Max a) Source # | |
Ord a => Monoid (Max a) Source # |
|
(Semiring a, Ord a) => Semiring (Max a) Source # | |
type Rep1 Max Source # | |
type Rep (Max a) Source # | |
The "Tropical" or min-plus semiring. It is a semiring where:
<+>
=min
zero
= ∞ -- represented byNothing
<.>
=<+>
one
=zero
Note that we can't use Min
from Semigroup
because annihilation needs to hold:
∞<+>
x = x<+>
∞ = ∞
Taking ∞ to be maxBound
would break the above law. Using Nothing
to represent it follows the law.
Monad Min Source # | |
Functor Min Source # | |
Applicative Min Source # | |
Foldable Min Source # | |
Generic1 Min Source # | |
Eq a => Eq (Min a) Source # | |
Ord a => Ord (Min a) Source # | |
Read a => Read (Min a) Source # | |
Show a => Show (Min a) Source # | |
Generic (Min a) Source # | |
Ord a => Semigroup (Min a) Source # | |
Ord a => Monoid (Min a) Source # |
|
(Semiring a, Ord a) => Semiring (Min a) Source # | |
type Rep1 Min Source # | |
type Rep (Min a) Source # | |