Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Messages: [ Previous | Next ]
 Victor Porton Posts: 621 Registered: 8/1/05
Re: Product, Filters and Quantales
Posted: Oct 17, 2013 1:18 PM

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.

> /\_(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}?

> 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}?

> K in (D xx F_{0}) o (D xx F_{(r,oo)})

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

Every set 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.

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