Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.



Re: Product, Filters and Quantales
Posted:
Oct 17, 2013 11:11 PM


On Thu, 17 Oct 2013, Victor Porton wrote: > William Elliot wrote: > > 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)})] ? > > (D xx F_{0}) o /\_(r>0) (D xx F_{(r,oo)}) = > (D xx F_{0}) o (D xx F_{(0,oo)}) = > D xx F_{0} != > 0 = > /\_(r>0) [(D xx F_{0}) o (D xx F_{(r,oo)})] > > So not. > Prove it.
> > /\_(r>0) (D xx F_{(r,oo)}) = D xx /\_(r>o) F_{(r,oo)} > > . . = D xx {R} > > /\_(r>0) (D xx F_{(r,oo)}) = > D xx (0;oo) = > D xx /\_(r>o) F_{(r,oo)} > > What is {R}? > { a,b } = { x  x = a or x = b } {a} = { x  x = a }
> > 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} > > What is {R}? > You don't know that notation?
> > K in (D xx F_{0}) o (D xx F_{(r,oo)}) > > (D xx F_{0}) o (D xx F_{(r,oo)}) = 0 > What's 0? Is it different than the 0 in R. If so, then don't use it; use some different notation.
> Every set K in (D xx F_{0}) o (D xx F_{(r,oo)})
Huh? That incomplete sentence fragment makes no sense.
> > . . 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. > >



