monoid-subclasses
=================

### Subclasses of Semigroup and Monoid with a solid theoretical foundation and practical purposes ###

The monoid-subclasses package has been released [on
Hackage](http://hackage.haskell.org/package/monoid-subclasses). The package defines several classes that are richer
than [semigroups](http://hackage.haskell.org/package/base/docs/Data-Semigroup.html#t:Semigroup) and
[monoids](http://hackage.haskell.org/package/base/docs/Data-Monoid.html#t:Monoid) but less demanding than
[groups](http://hackage.haskell.org/package/groups/docs/Data-Group.html):

* [Reductive](http://hackage.haskell.org/package/monoid-subclasses/docs/Data-Semigroup-Cancellative.html#t:Reductive)
provides the operator `</>` which acts as a partial inverse of the semigroup `<>` operator.
* [Cancellative](http://hackage.haskell.org/package/monoid-subclasses/docs/Data-Semigroup-Cancellative.html#t:Cancellative)
is a subclass of `Reductive` that provides additional guarantees about the `</>` operation result:

        (a <> b) </> a == Just b
        (a <> b) </> b == Just a

    Every group (*i.e.*, every `Monoid a` with the operation `inverse :: a -> a`) is a cancellative monoid where `a </> b = Just (a <> inverse b)` but not every `Cancellative` monoid is a group.
    
* [GCDMonoid](http://hackage.haskell.org/package/monoid-subclasses/docs/Data-Monoid-GCD.html#t:GCDMonoid) is a subclass of `Reductive` and `Monoid` that provides the `gcd` operation for getting the greatest common denominator for two given monoid values.
* [Monus](http://hackage.haskell.org/package/monoid-subclasses/docs/Data-Monus.html#t:Monus) provides the `<\>` monus operation. The set difference is one familiar instance of this operation.
* [MonoidNull](http://hackage.haskell.org/package/monoid-subclasses/docs/Data-Monoid-Null.html) class provides the Boolean `null` operation that checks if the argument monoid is `mempty`.
* [Factorial](http://hackage.haskell.org/package/monoid-subclasses/docs/Data-Semigroup-Factorial.html) and [FactorialMonoid](http://hackage.haskell.org/package/monoid-subclasses/docs/Data-Monoid-Factorial.html) classes represent semigroups and monoids that can be split up into irreducible factors.

That's the theoretical point of view. From the practical point of view, the main purpose of the _monoid-subclasses_ package is similar to that of [ListLike](http://hackage.haskell.org/package/ListLike/docs/Data-ListLike.html) - to provide unifying abstractions for various monoidal data types in Haskell, primarily [String](http://hackage.haskell.org/package/base/docs/Data-String.html#t:String), [ByteString](http://hackage.haskell.org/package/bytestring/docs/Data-ByteString.html#t:ByteString), and [Text](http://hackage.haskell.org/package/text). All three types are already instances of the [Monoid](http://hackage.haskell.org/package/base/docs/Data-Monoid.html#t:Monoid) class. While that abstraction is useful for building sequences of data, it doesn't help with deconstructing them.

That being said, there are two major differences in the goals of _ListLike_ and _monoid-subclasses_:
  * _ListLike_ strives to reproduce the standard [Data.List](http://hackage.haskell.org/package/base/docs/Data-List.html) interface, whereas _monoid-subclasses_ builds from deeper theoretical foundations; and
  * The _monoid-subclasses_ implementation uses standard Haskell 2010, with the exception of two minor extensions which can be worked around if necessary.

The [incremental-parser](http://hackage.haskell.org/package/incremental-parser) package can serve as a compact example
of a parser library that can be applied to different input types thanks to _monoid-subclasses_. There is also
[picoparsec](http://hackage.haskell.org/package/picoparsec), a fork of
[attoparsec](http://hackage.haskell.org/package/attoparsec), and the heavy-duty
[grammatical-parsers](http://hackage.haskell.org/package/grammatical-parsers) library.

A more thorough description of the library design can be found in the Haskell Symposium 2013 paper [Adding Structure
to Monoids ](https://github.com/blamario/monoid-subclasses/wiki/Files/HaskellSymposium2013.pdf)