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Re: Product, Filters and Quantales
Posted:
Oct 18, 2013 9:13 AM


William Elliot wrote:
> 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.
Which of the above equalities is unclear?
>> > /\_(r>0) (D xx F_{(r,oo)}) = D xx /\_(r>o) F_{(r,oo)} >> > . . = D xx {R}
Yes.
>> /\_(r>0) (D xx F_{(r,oo)}) = >> D xx (0;oo) = >> D xx /\_(r>o) F_{(r,oo)}
Yes.
>> 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}
No,
(D xx F_{0}) o /\_(r>0) (D xx F_{(r,oo)}) = D o F_{0}
>> > 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.
I meant 0 = PR
>> > . . 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.



