[Haskell-cafe] Category Theory woes

[Daniel Fischer]

Am Donnerstag 18 Februar 2010 21:47:02 schrieb 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.

Typo, iii. c(c(x)) = c(x), of course.

If we replace "set" by "lattice element" and "contains" by ">=", the 
definitions are equivalent. The one you quoted is better, though.

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