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

 Messages: [ Previous | Next ]
 fom Posts: 1,968 Registered: 12/4/12
Re: Prime Interger Topology
Posted: Oct 20, 2013 10:42 PM

On 10/20/2013 9:17 PM, William Elliot wrote:
> On Sun, 20 Oct 2013, fom wrote:
>

>> By the way, I just formulated an arithmetic based on
>> my set theoretic axioms and the fact that the natural
>> numbers enjoy a topology called the prime integer topology.

>
> 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?
>

I am not really applying it. I am constructing it. Or,
better, I am trying to construct as much of it as needed.

The integers loosely relate to the naturals via absolute
values. So, here is part of the expository for my
difference relations. Note that the difference relations
actually introduce the notion of a directed set by virtue
of the "commutativity" of what needs to be expressed,

In fact, it is the difference relation that introduces
a usable order...

<beginquote>

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.

This expresses,

8 - 3 = 5

8 - 5 = 3

in relation to

3 + 5 = 8

Observe, by this requirement in the definition of the
related relations, that the directed set structure is
imposed on the domain.

Difference relations are not simply "the opposite"

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.

<endquote>

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