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Re: Product, Filters and Quantales
Posted:
Oct 16, 2013 7:51 AM


William Elliot wrote:
> On Tue, 15 Oct 2013, Victor Porton wrote: > > If C subset P(S), then F(A) is the filter for S on P(S) generated by C. > If A subset S, then F_A = F{{A}) the principal filter generated by A > If F,G are filters, then F xx G = F({ AxB  A in F, B in G }). > > To recap from your errors and hard to use notation, is this the counter > example for > . . F o inf_k Gk = inf{ F o Gk  k in K } > where F and the Gk's are filters for products? > > D = F({ (r,r) subset R  0 < r }, the neighborhood filter for 0 in R. > F = D xx F_{0} is a filter for RxR on P(RxR). > >> Correction: >> G_e = up{0} x up(e;+oo) where "x" means reloidal product. > > Does G_r = D xx F_{(r,oo)}? > > Is this your counter example? > . . F o /\{ G_r  0 < r } /= /\{ F o G_r  0 < r }
Yes.
> PS. Don't forget for filters that > . . F o inf{ G_r  0 < r } = F o /\{ G_r  0 < r } > and > . . inf{ F o G_r  0 < r } = /\{ F o G_r  0 < r } > where /\ is great intersection.



