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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 counterexample 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 }



