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Topic: Which term to choose?
Replies: 41   Last Post: Nov 9, 2013 5:20 AM

 Messages: [ Previous | Next ]
 Victor Porton Posts: 621 Registered: 8/1/05
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 non-empty.
>>

> 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.

Date Subject Author
10/25/13 Victor Porton
10/25/13 Peter Percival
10/25/13 fom
10/25/13 William Elliot
10/26/13 William Elliot
10/26/13 Victor Porton
10/26/13 William Elliot
10/27/13 Victor Porton
10/27/13 William Elliot
10/28/13 Victor Porton
10/29/13 William Elliot
10/29/13 Victor Porton
10/30/13 William Elliot
10/30/13 Victor Porton
10/30/13 William Elliot
10/31/13 Victor Porton
11/1/13 William Elliot
11/1/13 Victor Porton
11/1/13 William Elliot
11/2/13 Victor Porton
11/2/13 William Elliot
11/3/13 Victor Porton
11/3/13 Victor Porton
11/3/13 William Elliot
11/4/13 William Elliot
11/4/13 Victor Porton
11/5/13 William Elliot
11/5/13 Victor Porton
11/6/13 William Elliot
11/6/13 Victor Porton
11/6/13 William Elliot
11/7/13 Victor Porton
11/7/13 William Elliot
11/8/13 William Elliot
11/8/13 Victor Porton
11/8/13 William Elliot
11/9/13 Victor Porton
11/9/13 William Elliot
11/9/13 William Elliot
11/9/13 Victor Porton
10/26/13 Victor Porton