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

 Messages: [ Previous | Next ]
 William Elliot Posts: 2,637 Registered: 1/8/12
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 non-empty.
> >

> 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, 2-x) | 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.
>
>

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