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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 20, 2013 3:59 AM

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)}

Victor's counter example.
. . F o /\{ G_r | 0 < r } /= /\{ F o G_r | 0 < r }

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)})] ?

Victor agrees.
/\_(r>0) (D xx F_{(r,oo)}) = D xx /\_(r>0) F_{(r,oo)} = D xx {R}

I state:
. . (D xx F_{0}) o /\_(r>0) (D xx F_{(r,oo)}) = D xx {R}
Victor Claims:
. . (D xx F_{0}) o /\_(r>0) (D xx F_{(r,oo)}) = D xx F_{0}

Using the above result where we agree, I prove my statement below.
Where is there an error in the proof?

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}

I continue to the final conclusion of equality.
Have you any quams about the ending statements?

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)}

(D xx F_{0}) o (D xx F_{(r,oo)}) = 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.

--
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.

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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