
Re: Product, Filters and Quantales
Posted:
Oct 17, 2013 3:41 AM


On Wed, 16 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). > > > > 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.
Does (D xx F_{0}) o /\_(r>0) (D xx F_{(r,oo)}) . . = /\_(r>0) [(D xx F_{0}) o (D xx F_{(r,oo)})] ?
/\_(r>0) (D xx F_{(r,oo)}) = D xx /\_(r>o) F_{(r,oo)} . . = D xx {R}
K in (D xx F_{0}) o (D xx {R} . . iff some A in DxxF_{0}, B in Dxx{R} } with AoB subset K . . iff some U in D, V in F_{0}, W in D with UxV o DxR subset K . . iff some U in D with UxR subset K iff K in D xx {R}
(D xx F_{0}) o /\_(r>0) (D xx F_{(r,oo)}) = D xx {R}
K in (D xx F_{0}) o (D xx F_{(r,oo)}) . . iff some A in D xx F_{0}, B in D xx F_{(r,oo)} with AoB subset K . . iff some U in D, V in F_{0}, W in D, X in F_{(r,oo)} . . . . with UxV o WxX subset K . . iff some U in D, X in F_{(r,oo)} with UxX subset K . . iff K in D xx F_{(r,oo)}
/\_(r>0) [(D xx F_{0}) o (D xx F_{(r,oo)})] . . = /\_(r>0) (D xx F_{(r,oo)}) = D xx /\_(r>0) F_{(r,oo)} = D xx {R}
Yes, they're equal.

