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