[Haskell-cafe] vector-space and standard API for vectors

[Jacques Carette]

  On 23/10/2010 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? 

People usually use additive notation for commutative monoids, and 
multiplicative notation for generic monoids.  It's a convention, nothing 
else.  Otherwise, of course they are isomorphic.

> 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.

How you do this really depends on what features you have for 
naming/renaming, how notions of inheritance/subtyping work, etc.

Having multiple names for structures which are, in fact, isomorphic, 
turns out to be really convenient when you want to combine the 
structures without duplicating declarations.  Let me stress that again: 
'really convenient'.  Not required, manditory, or anything else like that.

Of course, all we really require is any Turing complete language.  So if 
'require' is the main criterion, we should all still be programming in 
assembler.  There really is a reason we're having this discussion on the 
Haskell-cafe mailing list...

Jacques