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

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Victor Porton

Posts: 529
Registered: 8/1/05
Re: Product, Filters and Quantales
Posted: Oct 18, 2013 9:13 AM
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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.



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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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
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10/21/13
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10/20/13
Read Re: Product, Filters and Quantales
William Elliot

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