# semirings

two monoids as one, in holy haskimony http://github.com/chessai/semirings

Version on this page: | 0.1.3.0 |

LTS Haskell 11.22: | 0.1.3.0 |

Stackage Nightly 2018-12-10: | 0.2.1.1 |

Latest on Hackage: | 0.2.1.1 |

**chessai**

**chessai**

#### Module documentation for 0.1.3.0

# semirings

Haskellers are usually familiar with monoids and semigroups. A monoid has an appending operation `<>`

or `mappend`

and an identity element `mempty`

. A semigroup has an append `<>`

, but does not require an `mempty`

element.

A Semiring has two appending operations, ‘plus’ and ‘times’, and two respective identity elements, ‘zero’ and ‘one’.

More formally, A semiring R is a set equipped with two binary relations + and *, such that:

- (R, +) is a commutative monoid with identity element 0:
- (a + b) + c = a + (b + c)
- 0 + a = a + 0 = a
- a + b = b + a

- (R, *) is a monoid with identity element 1:
- (a * b) * c = a * (b * c)
- 1 * a = a * 1 = a

- Multiplication left and right distributes over addition
- a * (b + c) = (a * b) + (a * c)
- (a + b) * c = (a * c) + (b * c)

- Multiplication by ‘0’ annihilates R:
- 0 * a = a * 0 = 0

# *-semirings

A *-semiring (pron. “star-semiring”) is any semiring with an additional operation ‘star’ (read as “asteration”), such that:

- star a = 1 + a * star a = 1 + star a * a

A derived operation called “aplus” can be defined in terms of star by:

- star :: a -> a
- star a = 1 + aplus a
- aplus :: a -> a
- aplus a = a * star a

As such, a minimal instance of the typeclass ‘Star’ requires only ‘star’ or ‘aplus’ to be defined.

# use cases

semirings themselves are useful as a way to express that a type is both a commutative and associative monoid.

*-semirings are useful in a number of applications; such as matrix algebra, regular expressions, kleene algebras, graph theory, tropical algebra, dataflow analysis, power series, linear recurrence relations.

Some relevant (informal) reading material:

http://stedolan.net/research/semirings.pdf

http://r6.ca/blog/20110808T035622Z.html

https://byorgey.wordpress.com/2016/04/05/the-network-reliability-problem-and-star-semirings/

# additional credit

Some of the code in this library was lifted directly from the Haskell library ‘semiring-num’.

## Changes

## 0.2.1.1: [2018.XX.XX]

- Fixed build on GHC-7.4
- Provide
`Semiring`

and`Ring`

for an arbitrary`Num`

via`WrappedNum`

newtype.

## 0.2.1.0: [2018.09.26]

- Removed use of DefaultSignatures
- Removed free semiring

## 0.2.0.1: [2018.07.28]

- Add instances for
`Op`

,`Equivalence`

,`Comparison`

, and`Predicate`

from Data.Functor.Contravariant (upcoming base 4.12.0.0) - docfix for (prod -> product, prod’ -> product’)

## 0.2.0.0: [2018.07.23]

- Fixed the
`Semiring`

instances of`Set`

,`HashSet`

,`Vector`

,`Storable Vector`

,`Unboxed Vector`

. - Removed the
`Semiring`

instances of`Seq`

,`Alt`

,`Endo`

. - Added comprehensive test suite that tests all
`Semiring`

instances defined in Data.Semiring - Added Free semiring (Data.Semiring.Free)
- Added newtypes:
`Add`

,`Mul`

- Bounds for containers: [0.3,0.6] -> [0.5.4,0.6.0.9]
- Add semiring instance for
`Proxy`

- names changed: (prod -> product, prod’ -> product’)
- sum’ and product’ now use foldl’ instead of foldr’

## 0.1.2: [2018.05.04]

`semirings`

now builds back to GHC-7.4.1.- many doc fixes.

## 0.1.1: [2018.04.20]

- Remove unused
`coerce-util`

dependency.

## 0.1.0:

- Initial version.