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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: 524
Registered: 8/1/05
Re: Product, Filters and Quantales
Posted: Oct 19, 2013 10:24 AM
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William Elliot wrote:

> 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.


Sorry, typo. It should be:

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

> "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
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
William Elliot
10/21/13
Read Re: Prime Interger Topology
fom
10/20/13
Read Re: Product, Filters and Quantales
William Elliot

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