Am Donnerstag 07 Mai 2009 10:23:41 schrieb j.waldmann:

* in the Appendix on notation,  he defines composition of functions the "wrong way around" (i.e., the Haskell way, (f.g)(x) = f(g(x)). This is of course very awkward, especially considering that functions are relations, and an element of a relation is a directed edge, and there's a very obvious meaning of how edges are composed to give a path. Well, the book's "solution" is to avoid to define composition of relations! (p. 1155, they only define the n-fold composition of a relation with itself, and there the problem does not show.)

PS: I was shocked to find recently that Bourbaki (!) also defines functional composition "wrongly" (at least in the English edition of Elements of Math./Theory of Sets, II.3.7) but at least they are consistent and define composition of relations ("graphs") accordingly. They are very honest about this, e.g. "Let G1, G2, G3 be graphs. Then (G3 * G2) * G1 = G3 * (G2 * G1)."  (note the order of the variables)

That's the way composition of functions (and more generally relations) has been done for ages in mathematics. And since one writes f(x) and not x.f, it's the consistent way to define the composition. Of course, if centuries ago people had decided to write the argument before the function, composition would've been defined the other way round. They haven't.

Am Freitag 08 Mai 2009 13:23:09 schrieb Wolfgang Jeltsch:

Am Donnerstag, 7. Mai 2009 14:42 schrieb Daniel Fischer:

...

Of course, if centuries ago people had decided to write the argument before the function, composition would've been defined the other way round. They haven't.

Algebraists used to write x f instead of f(x) at least in the 1980s.

I think that should read *some* algebraists... Though I had no contact with algebraists in the 1980s, if that practice had been ubiquitous among them, I would have expected it to show up in the textbooks. I've never seen it in an algebra book, nor was it used in any algebra lecture I attended in the 1990s, so I doubt it was very widespread. Don't get me wrong, that notation does make sense and has some advantages over the conventional one, I wouldn't oppose a change, though it would take some time to get used to it. All I'm saying is that the overwhelming majority of mathematicians doesn't use it.

I think, also category theorists often wrote (write?) composition with the first morphism on the left, i.e., “the other way round”.

Yeah, I heard that, too. It's a field where the advantages of postfix notation show clearly and a young one, so for them it was relatively easy to switch. You'd have a hard time persuading the statisticians and analysts, though.

Best wishes, Wolfgang

Cheers, Daniel

j.waldmann wrote on 08.05.2009 17:39:

Wow. This is really a bikeshed discussion. My post contained praise for the book, a critique of one of its design decisions (intermediate language not explicitely typed), and a syntactical remark. Guess what the discussion is about. (This is also known as http://www.haskell.org/haskellwiki/Wadlers_Law)

If you're advocating (f.g) = \ x -> f(g(x)), then:

1. Define composition of relations.

R1 . R2 = {(x, y) | \exists z : (x, z) \in R2 & (z, y) \in R1} That's the definition I've usually seen. (If I'm not mistaken).

2. Is a function a special case of a relation?

Depends on the definition (i.e., some books define a relation on (A,B) as a subset in A\times B, and a function A -> B as a triple (A, B, G), where G is a subset in A\times B), but, generally speaking, yes.

Am Freitag, 8. Mai 2009 14:31 schrieb Daniel Fischer:

Though I had no contact with algebraists in the 1980s,

I also hadn’t. However, nowadays I have contact with someone who was an algebraist in the 1980s. It’s my boss (professor), by the way. :-)

...

I think, also category theorists often wrote (write?) composition with the first morphism on the left, i.e., “the other way round”.

Yeah, I heard that, too. It's a field where the advantages of postfix notation show clearly and a young one, so for them it was relatively easy to switch.

However, I fear that all those other mathematicians who define f . g = \x -> f(g(x), have made the category theorists switch to this suboptimal notation (first morphism on the right). At least, I cannot remember seeing the other notation (first morphism on the left) in category theory literature so far. It’s just that my above-mentioned professor told me that category theorists would use the first-morphism-on-the-left notation. Best wishes, Wolfgang

On Tue, May 12, 2009 at 1:04 PM, Max Rabkin wrote:

On Tue, May 12, 2009 at 1:41 PM, Wolfgang Jeltsch wrote:

...

At least, I cannot remember seeing the other notation (first morphism on the left) in category theory literature so far. It’s just that my above-mentioned professor told me that category theorists would use the first-morphism-on-the-left notation.

I've seen the notation f;g for g.f somewhere (and Wikipedia mentions it). I think it's less ambiguous than just fg (which I've seen for f.g too), but in Haskell we have the option of >>>. A flipped application might be nice to go with it. How about >$> ?

FYI: Unicode U+2A3E Zed notation relational composition (small circle over a 9) Examples at http://staff.washington.edu/jon/z/toolkit.html#pair3