On Dec 19, 2011, at 1:03 PM, Alexander Solla wrote:
The "incidental" comment is significantly more clear than an English description.
That is only true for someone who has already seen a sentence like that one before and so can immediately pick up what it is getting at. :-) In particular, if one has never heard of a semigroup then the sentence is not very helpful.
I would rather see commutative diagrams (or what amounts to the same, usage examples) that describe the behavior than a "plain English" description.
I find it amusing that anyone would consider commutative diagram to be the same thing as usage examples for anyone other than a mathematician. :-) Nonetheless, I see your point that examples may be clearer than English, so how about:
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This instance satisfies the property that, for all x any y:
(1) Nothing `mappend` Nothing = Nothing (2) Just x `mappend` Nothing = Just x
(3) Nothing `mappend` Just y = Just y (4) Just x `mappend` Just y = Just (x `mappend` y)
(Warning: Note that rule (4) for this instance is different from the case of the MonadPlus/Alternative instances where the Just y value is discarded so that Just x `mplus` Just y = Just x <|> Just y = Just x.)
Formally, this instance performs the standard procedure of turning an arbitrary semigroup into a monoid by simply adding a new element to the semigroup to serve as the identity element, where in this case the identity element is the Nothing value of Maybe; unfortunately, since the base libraries do not come with a Semigroup typeclass, this process is expressed in code as lifting from the Monoid typeclass.
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