[Haskell-cafe] Monad laws in presence of bottoms
wren ng thornton
wren at freegeek.org
Wed Feb 22 06:30:46 CET 2012
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
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