Dan Piponi wrote:
On Fri, Jan 16, 2009 at 12:09 PM, Paul Moore <p.f.mo...@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
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