[Haskell-beginners] Monoids and Groups

[Boris]

Hi,

While they have similarities, they are not the same. Semigroup (S,*) is a set S with associative binary operation *. Monoid is a semigroup that has identity element i such that i * a = a and a * i = a. And group is a monoid in which every element of set has it’s own inverse: a * b = i and b * a = i where b is an inverse of a.

Let’s look at the definition of Monoid in haskell.

class Monoid a where
        mempty  :: a
        -- ^ Identity of 'mappend'
        mappend :: a -> a -> a
        -- ^ An associative operation
        mconcat :: [a] -> a

        -- ^ Fold a list using the monoid.
        -- For most types, the default definition for 'mconcat' will be
        -- used, but the function is included in the class definition so
        -- that an optimized version can be provided for specific types.

        mconcat = foldr mappend mempty
So we have a set of a, associative binary operation mappend and identity element mempty. The only difference between Monoid a in haskell and monoid in algebra is that Monoid in haskell has mconcat function in it’s definition. But you can ignore it.

I hope it helps.

Cheers, d12frosted


On March 25, 2015 at 16:03:14, Shishir Srivastava (shishir.srivastava at gmail.com) wrote:

Hi, 

Reading about Monoids it seems they derive a lot on the algebraic structures of 'Groups' ? 

Is it then correct to assume that Monoids can be used to represent 'Groups' ?

If not are there any standard haskell libraries which represent algebraic structures such as 'Groups' , 'Fields' etc.

Thanks,
Shishir Srivastava

_______________________________________________  
Beginners mailing list  
Beginners at haskell.org  
http://mail.haskell.org/cgi-bin/mailman/listinfo/beginners  
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://mail.haskell.org/pipermail/beginners/attachments/20150326/ec0ef599/attachment.html>