
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 <=.

