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

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William Elliot

Posts: 1,521
Registered: 1/8/12
Re: Product, Filters and Quantales
Posted: Oct 17, 2013 11:11 PM
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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.

>
>



Date Subject Author
10/9/13
Read Product, Filters and Quantales
William Elliot
10/10/13
Read Re: Product, Filters and Quantales
Victor Porton
10/11/13
Read Re: Product, Filters and Quantales
William Elliot
10/11/13
Read Re: Product, Filters and Quantales
Victor Porton
10/12/13
Read Re: Product, Filters and Quantales
William Elliot
10/12/13
Read Re: Product, Filters and Quantales
Victor Porton
10/12/13
Read Re: Product, Filters and Quantales
William Elliot
10/14/13
Read Re: Product, Filters and Quantales
Victor Porton
10/15/13
Read Re: Product, Filters and Quantales
William Elliot
10/15/13
Read Re: Product, Filters and Quantales
Victor Porton
10/16/13
Read Product, Filters and Quantales
William Elliot
10/16/13
Read Re: Product, Filters and Quantales
Victor Porton
10/17/13
Read Re: Product, Filters and Quantales
William Elliot
10/17/13
Read Re: Product, Filters and Quantales
Victor Porton
10/17/13
Read Re: Product, Filters and Quantales
William Elliot
10/18/13
Read Re: Product, Filters and Quantales
Victor Porton
10/18/13
Read Re: Product, Filters and Quantales
William Elliot
10/19/13
Read Re: Product, Filters and Quantales
Victor Porton
10/19/13
Read Re: Product, Filters and Quantales
William Elliot
10/19/13
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William Elliot
10/20/13
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10/20/13
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William Elliot
10/20/13
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10/20/13
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William Elliot
10/20/13
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William Elliot
10/20/13
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10/20/13
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10/20/13
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10/21/13
Read Re: Prime Interger Topology
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10/21/13
Read Re: Prime Interger Topology
William Elliot
10/21/13
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fom
10/20/13
Read Re: Product, Filters and Quantales
William Elliot

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