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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: Prime Interger Topology
Posted: Oct 20, 2013 10:58 PM

On Sun, 20 Oct 2013, fom wrote:
> On 10/20/2013 9:17 PM, William Elliot wrote:
> >
> > The prime integer topology for N is the topology generated
> > by the subbase Bs = { (n + pN) /\ N | n in Z, p prime }
> >
> > What if the prime integer topology were applied to Z?

>
> The axiom for difference relations is given by,
>
> AwAxAyAz( -(w,x,y,z) <-> ( ( +(w,x,z,y) /\ -(w,y,x,z) ) /\ ( ~( -(w,y,z,x) )
> /\ ( ~( -(w,x,z,y) ) /\ ( ~( -(w,z,x,y) ) /\ ~( -(w,z,y,x) ) ) ) ) ) )
>
> The first conjunct in the expression,
> +(w,x,z,y) /\ -(w,y,x,z)
>
> asserts that there must exist an additive relation
> by which z is the sum of x and y. The second conjunct
> asserts that in relation to such a sum, the addends of
> the sum are commutative with respect to difference relations.

Why relations? Aren't functiosn used to manage the undue complexity of
relations?

> This expresses,
>
> 8 - 3 = 5
> 8 - 5 = 3
>
> in relation to
>
> 3 + 5 = 8
>
> The expression
>
> ( ~( -(w,y,z,x) ) /\ ( ~( -(w,x,z,y) ) /\ ( ~( -(w,z,x,y) ) /\ ~( -(w,z,y,x) )
> ) ) )
>
> shows the relationship of the difference relations to
> order on the domain. Since difference relations are understood
> with respect to sums, the sum of two terms may not occur in
> specific indices of the relation.
>
> That is, the fourth term in a difference relation is always greater
> than the second and third terms.

What a difficult way to define <=.

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