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Re: Product, Filters and Quantales
Posted:
Oct 18, 2013 10:41 PM


On Fri, 18 Oct 2013, Victor Porton wrote: > William Elliot 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)})] > > 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}
No, the composition of two reloids is a reloid. It can't be the composition of two filters for R on P(R) because composition isn't defined for filters except for filters for products which D and F_{0} certainly aren't.
> >> > 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 > "PR"? The acronym for "Public Relations"? Perhaps you mean P(R), the largest filter for R.
> >> > . . 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. >



