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Topic: Product, Filters and Quantales
Replies: 31   Last Post: Oct 21, 2013 7:52 AM

 Messages: [ Previous | Next ]
 Victor Porton Posts: 621 Registered: 8/1/05
Re: Product, Filters and Quantales
Posted: Oct 14, 2013 12:33 PM

William Elliot wrote:

> On Sat, 12 Oct 2013, Victor Porton wrote:
>> >
>> >> The following is a counter-example for
>> >>
>> >> F o inf_k Gk = inf{ F o Gk | k in K }
>> >>
>> >> Let D = Ft { (-e;e) | e>0 }
>> >>
>> >> ("Ft" means the filter generated by the given base, right?)

>> > D is a filter for R on P(R) ?
>> Yes.
>
> D is the neighborhood filter for 0 in R.
>

>> >> F = D x up{0}
>> >>

>> > up {0} = { r in R | 0 <= r } is not a filter. Do you mean
>> > up {{0}} = { A in P(R) | {0} subset A }?

>
> So which is it for up 0? The first or the second?
> BTW, the first is a filter on (R,<=).

Clearly the second.

up {0}

is the principal filter on R generated by the set {0}/

>> G_e = up{0} x (e;+oo)
>>

>> >> Then /\G = up{0} x up(e;+oo)
>>
> Which is correct?
> up(e,oo) = { x in R | e < 0 } is a filter on (R,<=)
> or
> up(e,oo) = { A in P(R) | (e,oo) subset A } a filter for R on P(R)?

Second.

I don't mess with filters on (R,<=), these are essentially outside of my
research topic.

>> G = { G_e | e>0 }
>>

>> >> up{0} x (e;+oo) = up{0} x up{0}
>> >>
>> >> So F o inf_k Gk != inf{ F o Gk | k in K }

Date Subject Author
10/9/13 William Elliot
10/10/13 Victor Porton
10/11/13 William Elliot
10/11/13 Victor Porton
10/12/13 William Elliot
10/12/13 Victor Porton
10/12/13 William Elliot
10/14/13 Victor Porton
10/15/13 William Elliot
10/15/13 Victor Porton
10/16/13 William Elliot
10/16/13 Victor Porton
10/17/13 William Elliot
10/17/13 Victor Porton
10/17/13 William Elliot
10/18/13 Victor Porton
10/18/13 William Elliot
10/19/13 Victor Porton
10/19/13 William Elliot
10/19/13 William Elliot
10/20/13 fom
10/20/13 William Elliot
10/20/13 fom
10/20/13 William Elliot
10/20/13 William Elliot
10/20/13 fom
10/20/13 William Elliot
10/20/13 fom
10/21/13 fom
10/21/13 William Elliot
10/21/13 fom
10/20/13 William Elliot