
Re: Partition of a filter
Posted:
Nov 9, 2013 5:20 AM


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.
I state this:
Definition 3.62: ... Strong partition of an element a in A is a set S in PA\{0} such that...
Note "\{0}".
> 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)).
U,V disjoint iff bottom = (sup U) inf (sup V)
Not /= but =.
>> 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.
No, I say they are not the same.
Example 4.247. There exists a weak partition which is not a strong partition.
> 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.
No.
{a} is a weak partition of a. Why have you concluded it isn't?
>> > 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.
In the PDF file I have sent you 4.153 is a theorem unrelated with ultrafilters.
>> > 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.
William, REVERSE order.
>> > 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.
It should be "torn" instead. I will correct my manuscripts.
P.S. Please write in an other thread, my news reader is overflowed.

