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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 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} = { 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 }



