[Haskell-cafe] Join and it's relation to >>= and return

[Benjamin Franksen]

On Wednesday 09 June 2004 17:20, Ron de Bruijn wrote:
> --- "Iavor S. Diatchki" <diatchki at cse.ogi.edu> wrote:
> Only I still find it weird that join is called a
> multiplication, because according to the definition of
> multiplication, there should be an inverse. I think,
> thus that multiplication is only defined on a group.
> And now still remains: why do they call it a
> multiplication, while by definition it's not. Or
> should I understand it as: there's a concept called
> multiplication and for different structures there's a
> definition? I think, now I think over it, that it
> would seem logical.
>
> It could be possible that the definition is incorrect,
> though. Does anyone knows of a definition that is more
> general (and not absolute nonsens ;))?

The term "multiplication" as it stands (i.e. without context) is not a 
defined mathematical concept. I.e. there is no (generally accepted) 
definition.

Of course multiplication of numbers, vectors, matrices, functions etc... 
are all well defined. Multiplication isn't even constrained to the 
operation on a semi-group as the example of multiplication of scalars 
to vectors shows.

You probably would not use the term "multiplication" for anything that 
is not at least a function f: A, B -> C where A, B, and C are sets (or 
at least classes). If A=B, then you would probably assume 
associativity, though there migth be "counter-examples" even for this 
rule.

Ben