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

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 Victor Porton Posts: 614 Registered: 8/1/05
Re: Partition of a filter
Posted: Nov 9, 2013 1:19 AM

William Elliot wrote:

> On Fri, 8 Nov 2013, Victor Porton wrote:
>> > Are these correct?
>> >
>> > Definition 3.60.
>> > A thorning of a point a in a complete lattice L
>> > is a subset A, of L\bottom with sup S = a and
>> > for all x,y in A, some d in L\bottom with d <= x,y

>>
>> What is "L\bottom with sup S"?
>>

> 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 an element a in a complete lattice L
>> > is a subset A, of L\bottom with sup A = a and for all
>> > x in A, there's some d in L\bottom with d <= a, sup A\x

>
> 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.

Every weak partition is a strong partition. Consequently strong partitions
exist.

> Where in the heck is Conjecture 4.153? In what section?

>> I mean that 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.

> Why the names thorning and partition for
> defintions that are unrelated to the words.

"Thorn" means to roughly thorn without proper "boundaries" unlike
partitions.

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