
Re: Partition of a filter
Posted:
Nov 9, 2013 3:42 AM


On Fri, 8 Nov 2013, William Elliot wrote: > 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 thornings. > 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.
Here's an example of a weak portition that's not a thorning. Let L = [0,2]^2 with coordinate wise ordering. A = { (x, 2x)  x in [0,2] } is a weak partiton of (2,2) because for all p in A, sup A\p = (2,2).
It's not a thorning because (0,2) and (2,0) in A. For the same reason, it's not a strong partition.
> > > 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 thorning of F by ultrafilters can have only one element > and there are no thornings 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 thorns, rose bush thorns. > There is no verb thorning. Thorns and the definition > of thorning are grossly mismatched. > >

