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Topic: Product, Filters and Quantales
Replies: 31   Last Post: Oct 21, 2013 7:52 AM

 Messages: [ Previous | Next ]
 William Elliot Posts: 2,637 Registered: 1/8/12
Re: Product, Filters and Quantales
Posted: Oct 19, 2013 10:59 PM

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.

--
On Sat, 19 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 }).

If F and the Gk's are filters for products, then
. . F o inf_k Gk = inf{ F o Gk | k in K }

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).
G_r = D xx F_{(r,oo)}

. . 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)})] ?

> >> 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.

> >> >> > 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}

>> (D xx F_{0}) o /\_(r>0) (D xx F_{(r,oo)}) = D xx F_{0}

Here we differ.

> >> >> > 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.

>
> I 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.

> >>
>
>

Date Subject Author
10/9/13 William Elliot
10/10/13 Victor Porton
10/11/13 William Elliot
10/11/13 Victor Porton
10/12/13 William Elliot
10/12/13 Victor Porton
10/12/13 William Elliot
10/14/13 Victor Porton
10/15/13 William Elliot
10/15/13 Victor Porton
10/16/13 William Elliot
10/16/13 Victor Porton
10/17/13 William Elliot
10/17/13 Victor Porton
10/17/13 William Elliot
10/18/13 Victor Porton
10/18/13 William Elliot
10/19/13 Victor Porton
10/19/13 William Elliot
10/19/13 William Elliot
10/20/13 fom
10/20/13 William Elliot
10/20/13 fom
10/20/13 William Elliot
10/20/13 William Elliot
10/20/13 fom
10/20/13 William Elliot
10/20/13 fom
10/21/13 fom
10/21/13 William Elliot
10/21/13 fom
10/20/13 William Elliot