[Haskell-cafe] Documentation [Comments from OCaml Hacker Brian Hurt]

[Cory Knapp]

Dan Piponi wrote:
> On Fri, Jan 16, 2009 at 12:09 PM, Paul Moore <p.f.moore at gmail.com> wrote:
>
>   
>> How about "associativity means that how you pair up the operations
>> does not affect the result"?
>>     
>
> I think a better way is this: If you have an element of a monoid, a,
> there are two ways to combine it with another element, on the left or
> on the right. You get
>
> a `mappend` x
> or
> x `mappend` a.
>
> Now suppose you're going to combine a with x on the left and y on the
> right. Associativity means it doesn't matter which you do first.
>
> You can think of each element of a monoid as having two sides. The
> idea is that the left side and right side are independent things that
> don't interfere with each other. For example, adding some stuff at the
> beginning of a list, and adding some stuff at the end of a list, don't
> affect each other, and it doesn't matter which you do first.
>
> That's the idea that associativity captures.
> --
> Dan
>   
Indeed, that's the idea that associativity captures; but explicitly 
pointing out that the left and the right side are their own bubbles is a 
bit misleading: addition is associative, but there is no "left and 
right." I think a better wording is:

"
If you have an element of a monoid, a, there are two ways to combine it 
with another element, on the left or on the right. You get

a `mappend` x
or
x `mappend` a.

Now suppose you're going to combine a with x on the left and y on the
right. Associativity means it doesn't matter which you do first.

So
x `mappend` (a `mappend` y) = (x `mappend` a) `mappend` y, but as we've 
pointed out,
a `mappend` x
is not necessarily the same as
x `mappend` a
although a specific monoid might have them be equal, for example Int 
(where mappend is *).


Is that better?

Cory