Re: [Haskell-cafe] Could someone teach me why we use Data.Monoid?

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 :

I see. Then what is about Dual and Endo? Especially Endo, I completely confused....

2009/11/14 Eugene Kirpichov :

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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 :

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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

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-- 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