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 ]
 William Elliot Posts: 2,637 Registered: 1/8/12
Re: Product, Filters and Quantales
Posted: Oct 18, 2013 10:41 PM

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.

> >> > 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.

> >> > . . 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