[Haskell-cafe] Category Theory woes

[Hans Aberg]

On 18 Feb 2010, at 20:20, Daniel Fischer wrote:

>> + definition backtracking: «A closure operation c is defined by the
>> property c(c(x)) = c(x).
>
> Actually, that's incomplete, ...

That's right, it is just the idempotency relation.

> ...missing are
> - c(x) contains x
> - c(x) is minimal among the sets containing x with y = c(y).

It suffices*) with a lattice L with relation <= (inclusion in the case  
of sets) satifying
   i. x <= y implies c(x) <= c(y)
  ii. x <= c(x) for all x in L.
iii. c(c(x)) = x.

   Hans

*) The definition in a book on lattice theory by Balbes & Dwinger.