
Re: Partition of a filter
Posted:
Nov 8, 2013 9:41 PM


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

