
Re: Partition of a filter
Posted:
Nov 9, 2013 2:37 AM


On Sat, 9 Nov 2013, Victor Porton wrote:
> >> > Are these correct? > > Definition 3.60. > > > A thorning of a point a in a complete lattice L > > is some A subset L\bottom for which sup A = a > > and for all x,y in A, bottom /= x inf y. > > (bottom is minimum element of L; S\a used for S\{a}) > > > > {a} is a thorning of a.
> > Definition 3.61.
> > A weak partition of a point a in a complete lattice L > > is some A subset L\bottom for which sup A = a > > and for all x in A, bottom /= x inf (sup A\x). > > > > {a} is not a weak partition of a. > > > >> > Definition 3.62. > > Correct. > > > A strong partition of a point a in a complete lattice L > > is some A subset L\bottom for which sup A = a and > > for all U,V subset S, (U,V disjoint iff bottom /= (sup U) inf (sup V)). > > > > There are no strong partitions because if U = A and V is empty > > bottom = (sup U) inf (sup V). > > U and V must be nonempty. > You need to state that in the definition.
A strong partition of a point a in a complete lattice L is some A subset L\bottom for which sup A = a and for all not empty U,V subset S, (U,V disjoint iff bottom /= (sup U) inf (sup V)).
> Every weak partition is a strong partition. Consequently strong partitions > exist.
According to Obvious 3.63 Strong partitions are weak partitions and weak partitions are thorning.
Thus by what you say, strong and weak partitions are the same.
What's obvious to me is that strong partitions are thronings. Do you have a proof that weak partitions are thornings or strong partitions? Strong partitions are not weak partitions for {a} (as above) is a strong partition of a. It is not a weak partition of a.
> > Where in the heck is Conjecture 4.153? In what section? > > I don't understand your order. Where, in your text, do I find Conjecture 4.153, that a filter can be partitioned into ultrafilters in the REVERSE order.
> > There is no thorning of a filter by ultrafilters. > The set of all ultrafilters below a filter (in the reverse order) is a > thorning of this filter.
If G,H are distinct ultrafilters for S, filter F subset G,H, then G sup H = P(S).
Reversing that, for all distinct ultrafilters G,H <= F, G inf H = 0. Thus a throning of F by ultrafilters can have only one element and there are no thronings for filters that aren't ultra.
> > Why the names thorning and partition for > > defintions that are unrelated to the words. > > "Thorn" means to roughly thorn without proper "boundaries" unlike > partitions. Makes no sense. Partitions are dividing into parts and there's no sense of that in your definitions.
A thorn is pointed part of a plant designed to scrach or prick. For example, blackberry throns, rose bush throns. There is no verb thorning. Throns and the definition of throning are grossly mismatched.

