For every monoid (M, *, u), the dual to it is the monoid (Dual M, \x y
-> y * x, u)
For every type A, there exists the A-endomorphism monoid (A->A, (.),
id). Endo A is just a newtype for A -> A.
More simply, dualization is flipping the binary operation, and the
endo monoid is the monoid of functions a->a with composition.
2009/11/13 Magicloud Magiclouds <magicloud.magiclo...@gmail.com>:
> I see. Then what is about Dual and Endo? Especially Endo, I completely
> confused....
>
> 2009/11/14 Eugene Kirpichov <ekirpic...@gmail.com>:
>> There is an astonishing number of things in programming that are monoids:
>> - Numbers, addition, 0
>> - Numbers, multiplication, 1
>> - Lists, concatenation, [] (including strings)
>> - Sorted lists, merge with respect to a linear order, []
>> - Sets, union, {}
>> - Maps, left-biased or right-biased union, {}
>> - Maps K->V, union where Vs for same K get merged in some other monoid, {}
>> - For any M: Subsets of M, intersection, M
>> - Any lattice with an upper bound, minimum, upper bound;
>> symmetrically for a lower-bounded set
>> - If (S, *, u) is a monoid, then (A -> S, \f g x -> f x * g x, \x ->
>> u) is a monoid
>> - Product (a,b) and co-product (Either) of monoids
>> - Parsers, alternation, a parser that always fails
>> - etc.
>>
>> The benefits of calling something a monoid arise from using
>> general-purpose structures operating on monoids:
>> - Finger trees http://apfelmus.nfshost.com/monoid-fingertree.html
>> - Aforementioned maps which merge values for a key in a given monoid
>> - Aforementioned monoids lifted to functions
>> - Monoidal folds (Data.Foldable)
>> - ...
>>
>> 2009/11/13 Magicloud Magiclouds <magicloud.magiclo...@gmail.com>:
>>> Hi,
>>> I have looked the concept of monoid and something related, but
>>> still, I do not know why we use it?
>>>
>>> --
>>> 竹密岂妨流水过
>>> 山高哪阻野云飞
>>>
>>> _______________________________________________
>>> Haskell-Cafe mailing list
>>> Haskell-Cafe@haskell.org
>>> http://www.haskell.org/mailman/listinfo/haskell-cafe
>>>
>>>
>>
>>
>>
>> --
>> Eugene Kirpichov
>> Web IR developer, market.yandex.ru
>>
>
>
>
> --
> 竹密岂妨流水过
> 山高哪阻野云飞
>
--
Eugene Kirpichov
Web IR developer, market.yandex.ru
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