Iavor S. Diatchki wrote:

Ron de Bruijn wrote:

...

I am pretty sure, that >>= is to monads what * is to for example natural numbers, but I don't know what the inverse of >>= is. And I can't really find it anywhere on the web(papers, websites, not a single sole does mention it.

this is not quie correct. (join & return) for a monad are like (*,1) or (+,0) for the set of integers. however those operations on integers have more structure than join and return.

there is no operation for "inverse". in mathematical terms: monads are a monoid (given that the notion is generalized considerably from its usual use), and not a group. if one was to add such an operation (i am not sure what it would do), but it would be of type: inverse :: M a -> M a (and of course must satisfy some laws)

I have *nothing* to add, just a question. Do you /anybody/ know of any edible work on ADJUNCTIONS in the context of Haskell structures? Perhaps instead of searching for 'inverses' one should think more about adjoints?... Jerzy Karczmarczuk

--- "Iavor S. Diatchki" wrote:

hi ron,

here are the relations between the two formulations of monads: (using haskell notation)

map f m = m >>= (return . f) join m = m >>= id

m >>= f = join (fmap f m)

there are quite a few general concepts that you need to understand in what sense monads are monoids, but to understand how monads work you don't need to know that.

Ron de Bruijn wrote:

...

I am pretty sure, that >>= is to monads what * is to for example natural numbers, but I don't know what the inverse of >>= is. And I can't really find it anywhere on the web(papers, websites, not a single sole does mention it.

this is not quie correct. (join & return) for a monad are like (*,1) or (+,0) for the set of integers. however those operations on integers have more structure than join and return.

there is no operation for "inverse". in mathematical terms: monads are a monoid (given that the notion is generalized considerably from its usual use), and not a group. if one was to add such an operation (i am not sure what it would do), but it would be of type: inverse :: M a -> M a (and of course must satisfy some laws)

also while you are pondering these things, it may be useful to use the "backward join" (=<<) :: (a -> m b) -> (m a -> m b). the reason for that is that strictly speaking tha arrow in the middle is different from the left and the right arrows

i am not sure how useful this information is for you, but if you have more questions ask away -iavor I found out what a group is: A group is a monoid each of whose elements is invertible.

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 information you give me is *very* usefull, because I don't just want to work with monads, I truly want to understand them. Ron __________________________________ Do you Yahoo!? Friends. Fun. Try the all-new Yahoo! Messenger. http://messenger.yahoo.com/

G'day all. Quoting Rik van Ginneken :

It is more even subtle if one considers the rotation group. The unit is keeping an object on its place. The multiplication is doing rotations sequently. Allright, one has an inverse here, but the rule (a*b)*c = a*(b*c) doesn't occur; try it with a banana or an apple!

Wrong. Rotations do indeed form a group. In three dimensions, it's isomorphic to the unit quaternion group. I think you're thinking of unit octonions, which don't form a group because octonion multiplication is not associative in general.

Please make a distinction betwixt "monoids" and "monads". The first is a group without an invertion.

...or a semigroup with an identity.

The latter is a construction in category theory (=abstract nonsense),

Them's fighting words! Cheers, Andrew Bromage

On Wednesday 09 June 2004 17:20, Ron de Bruijn wrote:

--- "Iavor S. Diatchki" 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