[Haskell-cafe] Monad laws in presence of bottoms

[wren ng thornton]

On 2/21/12 11:27 AM, MigMit wrote:
> Ehm... why exactly don't domain products form domains?

One important property of domains[1] is that they have a unique bottom 
element. Given domains A and B, let us denote the domain product as:

     (A,B) def= { (a,b) | a <- A, b <- B }

Which will inherit an ordering in the obvious/free way from the domain 
orderings on A and B. Since both A and B are domains, they have bottom 
elements:

     exists a0:A. forall a:A. (a0  <=_A  a)
     exists b0:B. forall b:B. (b0  <=_B  b)

However, there is no free ordering on:

     { (a0,b) | b <- B } \cup { (a,b0) | a <- A }

So all of those are minimal elements of (A,B) but none of them is a 
unique minimum; hence (A,B) is not a domain.

The smash product gets around this because it takes all those elements 
and makes them equal, just like a strict tuple would in Haskell.


[1] This is in the sense of domain theory. It has nothing (per se) to do 
with the many other uses of the term "domain" in mathematics.

-- 
Live well,
~wren