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Topic: Please nominate me for Abel Prize
Replies: 33   Last Post: Jul 19, 2013 10:22 PM

 Messages: [ Previous | Next ]
 William Elliot Posts: 2,637 Registered: 1/8/12
Re: Please nominate me for Abel Prize
Posted: Jul 17, 2013 11:50 PM

On Wed, 17 Jul 2013, Victor Porton wrote:
> William Elliot wrote:
> >
> >> Reloid is basically just a filter on a Cartesian product of two sets.
>
> The exact definition:
>
> A reloid is a triple (A;B;F) where A and B are sets and F is a filter on
> their product AxB.
>
> (Note that I allow a filter to be improper.)

What's an improper filter?

(A,B,F) is a reloid when F is a filter for AxB or an improper filter for AxB.

> A funcoid is a quadruple \$(A;B;a;b)\$ where \$A\$ and \$B\$ are sets, \$a\$ is a
> function from the set of filters on \$A\$ to the set of filters on \$B\$, \$b\$
> is a function from the set of filters on \$B\$ to the set of filters on \$A\$,
> subject to the following condition:

Skip the TeX is these ascii only newsgroup; it make reading harder.

Let FF(A) = { F | F filter over A }. Are improper filters included?

> For every filter X on A and every filter Y on B, the filter (on B) generated
> by the union of Y and a(X) is not the entire power set P(B) iff the filter
> (on A) generated by the union of X and b(Y) is not the entire power set
> P(A).

If S is a set, B subset P(S), then
filter< B > is the filter for S generated by B.

(A,B,f,g) is a funcoid when f:FF(A) -> FF(B), g:FF(B) -> FF(A),
for all X in FF(A), Y in FF(B),
filter< Y \/ f(X) > proper subset P(B)
iff fllter< X \/ g(Y) proper subset P(A)

(A,B,f,g) is a funcoid when f:FF(A) -> FF(B), g:FF(B) -> FF(A),
for all X in FF(A), Y in FF(B),
filter< Y \/ f(X) > = P(B) iff fllter< X \/ g(Y) > = P(A).
Is that correct?

> See http://nlab.mathforge.org/nlab/show/funcoid for a more understandable
> definition of funcoids than one in the paragraph above of this post.

No thanks, I've not the patience to mess with pdf stuff.