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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 15, 2013 7:34 AM

William Elliot wrote:

> On Mon, 14 Oct 2013, Victor Porton wrote:
>> William Elliot 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}} = { A in P(R) | {0} subset A }?
>> Clearly the second.
>>
>> up {0} is the principal filter on R generated by the set {0}
>>

>> >> G_e = up{0} x (e;+oo)
>
> Error: G_e is not a reloid for P(R) x P(R).

Correction:

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

where "x" means reloidal product.

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

>> >> >> up{0} x (e;+oo) = up{0} x up{0}
>
> Another error. What the hence do you mean?
> up{{0}} x up{ (e,oo) | 0 < e } ?

Reloidal product.

It should be up{0} not up {{0}}.

> If you do, then a mistake for up{ (e,oo) | 0 < e } /= 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