
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 nonempty.
Every weak partition is a strong partition. Consequently strong partitions exist.
> Where in the heck is Conjecture 4.153? In what section?
I don't understand your order.
>> 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.

