On 10/23/10 5:19 PM, Daniel Peebles wrote:
Just out of curiosity, why do you (and many others I've seen with similar proposals) talk about additive monoids? are they somehow fundamentally different from multiplicative monoids? Or is it just a matter of notation? When I was playing with building an algebraic hierarchy, I picked a "neutral" operator for my monoids (I actually started at magma, but it's the same thing) and then introduced the addition and multiplication distinction at semirings, as it seemed pointless to distinguish them until you have a notion of a distributive law between the two.
Mathematically speaking, of course you're right about the distinction only mattering once you get to semirings et al. But problems arise re the type class system and how to use it.
For example, in many contexts we just want a monoid and we don't care what kind it is (e.g., fun with fingertrees). For these, it's fine to use a neutral monoid with newtype wrappers to capture the multiple monoidal structures on certain types.
But in many other contexts we want to have multiple monoids in scope at once (e.g., when dealing with semirings,...). For these, the newtype approach is a nightmare of illegibility, so we should have (at least) two classes ((+),0) and ((*),1).
The question then becomes: which of these is the "primary" situation? If neutral monoids are (as H98 and H2010 assume), then one of the two classes for semirings could be the same as the neutral class (as in Ed's monads). But then, how should we decide whether the additive or multiplicative structure is more "neutral"? And why should there be this syntactic imbalance when, mathematically, there isn't?
I'd argue that neither usage is "primary", and therefore the best solution is actually to have three classes (neutral, additive, and multiplicative) with two functors (additive->neutral, multiplicative->neutral) to connect them[1]. In addition to removing the need to argue for primacy, it also removes the need to remember which monoid is considered the "neutral" one, as well as removing the syntactic imbalance between them. Both multiplicative and additive monoids would be on the base type, whereas the two neutral monoids would be newtypes.
[1] the Exp/Log isomorphism between additive and multiplicative are not considered here, as there are good arguments both for using newtypes and for not using them.
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