Maximum and Minimum monoids

[Herbert Valerio Riedel]

Gabriel Gonzalez <gabriel439 at gmail.com> writes:

[...]

> Also, it's not clear why something needs to be bounded to have a
> maximum.  Real numbers are unbounded, yet you can still take a maximum
> of a set of real numbers.

I just realized what confuses me about this bounded maximum/minimum
definitions: In the math literature I've been exposed to so far (see
also [1][2]), to satisify the definition of a maximal/minimal element of
a set that said element has to be actually contained in that set.

So it seems very confusing to me to call the element resulting from

  (Ord a, Bounded a) => Monoid (Max a)

with 'Max a' isomorphic to 'a' a proper "maximum" (as it violates the
definition for 'mempty'); on the other hand, the term "supremum"[3]
seems to match the semantics of the Monoid above better.

 [1]: http://en.wikipedia.org/wiki/Maximal_element#Definition
 [2]: http://mathworld.wolfram.com/Maximum.html
 [3]: http://en.wikipedia.org/wiki/Supremum

cheers,
  hvr